This is explained in more detail in the book The Science of Interstellar by physicist Kip Thorne, who came up with the original script treatment and was a consultant on the movie. The explanation is a bit involved, but feel free to skip to the last paragraph of this answer for a short summary.
The main idea about new physics in the movie was that gravity was affected by events happening in a higher spatial dimension, "the bulk". This idea was based on some real theoretical physics models involving branes (higher-dimensional analogues of strings in string theory), specifically models involving a large extra dimension where our 3D universe is a 3-brane sitting in a higher space dimension (which physicists do refer to as 'the bulk'). As Thorne explains in chapter 23, these models suggest that while most particles emitted in our 3-brane stay confined to it, gravitons can escape. If they were able to radiate off arbitrarily far away, that would mean the strength of the gravitational force would fall off with the cube of distance rather than the square as we observe. But in the Randall-Sundrum model, there is a type of strong spacetime warping, "Anti-deSitter warping", in the region of the bulk near our 3-brane, known as the "AdS layer", which causes most gravitons leaving our 3-brane to travel on bent paths that are confined to this region. This confinement means that gravity does continue to obey an inverse-square law at large distances (by an argument I won't try to reproduce here, but see either chapter 23 of Thornes' book or Lisa Randall's own book Warped Passages for more info), though at very short distances it could deviate from this. And because warping of spacetime in the AdS layer affects the behavior of gravity in our 3-brane, apparently this means that fluctuations in this warping could change the behavior of gravity. As Thorne says in this chapter:
By confining gravity, the AdS layer regulates its strength. In
Interstellar we see gravity's strength fluctuate, perhaps due to
fluctuations in the AdS layer. These fluctuations—gravitational
anomalies—play a huge role in Interstellar.
Thorne goes on to say in chapter 24 that in Intersellar's universe "gravitational anomalies" have been found in the gravitational tidal forces surrounding the Earth, and in chapter 25 he goes into detail about Professor Brand's proposed explanation for these anomalies, involving higher-dimensonal sources in the bulk, according to his own "science extrapolation" of what was going on in the movie (even if this wasn't spelled out onscreen):
In my extrapolation, it does not take long for the Professor to
convince himself that the anomalies are due to gravity from the fifth
dimension. From the bulk. Why?
The sudden changes in tidal gravity have no apparent source in our
four-dimensional universe. For example, in my extrapolation the
Professor's team sees the tidal gravity above an oil deposit switch,
in just a few minutes, from the pattern we expect (top picture in
Figure 25.2) to a radically different pattern (bottom picture). The
oil has not moved. The rocks have not shifted. Nothing in our
four-dimensional universe has changed except the tidal gravity. These
sudden changes must have a source. If the source is not in our
universe, on our brane, then there is only one other place it can be,
the Professor reasons: in the bulk.
He goes on to say that for technical reasons the Professor rejects the idea that it's just a matter of a higher-dimensional object in the bulk passing near our brane and exerting gravitational forces on it. Instead, the Professor concludes that the most likely explanation is that there are "bulk fields" which consist of collections of lines of force ('What kind of force lines, the Professor doesn't know') that generate tidal gravity when they pass through our brane. Thorne also says if our universe has only 3 spatial dimensions, then relativity says the gravitational constant G must be constant, but "if the bulk does exist, then the relativistic laws allow this G to change. It might, the Professor speculates, be controlled by the bulk fields. It probably is controlled by the bulk fields, he thinks." One of the anomalies Thorne imagines in the world of the film was a small but measurable change in the relative strength of the Earth's gravitational pull at different points on its surface, and the Professor thinks the best explanation for this was not just a change in tidal forces, but an actual variation in the value of G at different points in the space occupied by the Earth.
The equations on the Professor's blackboard were meant be "a mathematical description of the bulk fields and how they might generate anomalies" and control the value of G (along with two other applications he discusses: holding the wormhole open, and protecting our universe from collisions with other 'confining' branes). Note that he was originally not trying to find a full theory of quantum gravity, see the part I bolded in the quote below:
In creating this mathematics, the Professor was guided by the trove of
observational data his team was collecting (Chapter 24), and by
Einstein's relativistic laws of physics in five dimensions.
The Professor embodied all his insights in a single equation, THE
equation, which he wrote on one of the sixteen blackboards in his
office (Figure 25.7). Cooper sees the equation on his first visit to
NASA, and the equation is still there thirty years later, when Murph
has grown up to become a brilliant physicist in her own right, and is
helping the Professor try to solve it.
This equation is called an "Action." There is a well-known (to
physicists) mathematical procedure to begin with such an Action, and
from it deduce all the nonquantum physical laws. The Professor's
equation, in effect, is the mother of all nonquantum laws. But for it
to give birth to the right laws—the laws that predict correctly
how the anomalies are produced, how the wormhole his held open, how G
is controlled, and how our universe is protected—the equation must
have precisely the correct mathematical form. The Professor doesn't
know the correct form. He is guessing. His is an educated guess, but a
guess nevertheless.
His equation contains lots of guessing: guesses for things called
"U(Q), H_ij (Q^2), W_ij, and M (standard model fields)" on his
blackboard (Figure 25.7). In effect, these are guesses for the nature
of the bulk fields' force lines, and how they influence our brane, and
how fields in our brane influence them.
...
When the Professor and his team speak of "solving the equation," in my
extrapolation they mean two things. First, figure out the right
forms for all these things they are guessing: "U(Q), H_ij(Q^2), W_ij,
and M (standard model fields)." Second (following the well-known
procedure), deduce, from his equation, everything he wants to know
about our universe, about the anomalies, and most important, about how
to control the anomalies so as to lift colonies off the Earth.
When characters in the movie speak of "solving gravity," they mean the
same thing.
In the movie, when the Professor is very old, we see him and grown-up
Murph trying to solve his equation by iterations. On a blackboard,
they make a list of guesses for the unknown things (guesses that I
wrote on the board just before the scene was filmed; Figures 25.8 and
25.9). Then, in my extrapolation, Murph inserted each guess into a huge computer program they've written. The program computes the
physical laws for that guess, and those laws' predictions for how the
gravitational anomalies behave.
In my extrapolation, none of the guesses predicts anomalies that look
anything like the observations. But in the movie, the Professor and
Murph keep trying. They keep iterating: making a guess, computing the
consequences, abandoning the guess, and going on to the next guess,
one guess after another after another, until exhaustion sets in. Then
they begin again the next day.
A bit later in the movie, when the Professor is on his deathbed, he
confesses to Murph: "I lied, Murph. I lied to you." It is a poignant
scene. Murph infers that he knew something was wrong with his
equation, knew from the outset. And Dr. Mann tells the Professor's
daughter as much in an equally poignant scene on Mann's planet.
But in fact—Murph realizes, soon after the Professor's death—"His
solution was correct. He'd had it for years. It's half the answer."
The other half can be found inside a black hole. In a black hole's
singularity.
In chapter 25, Thorn explains how, in the movie, they could solve the equation if they did have a full theory of quantum gravity--they could discard the "quantum fluctuations" and be left with the non-quantum theory of the higher-dimensional fields that the Professor was looking for. They hope that data from the black hole could help them find a theory of quantum gravity:
If Professor Brand could discover the quantum gravity laws for the bulk as well as our brane, they by discarding those laws'
fluctuations, he could deduce the precise form of his equation
(Chapter 25). And that precise form would tell him the origin of the
gravitational anomalies and how to control the anomalies—how to employ
them (he hopes) to lift colonies off Earth.
In my extrapolation of the movie, the Professor knows this. And he
also knows a place where the quantum gravity laws can be learned:
inside singularities.
In real life, most physicists expect that a theory of quantum gravity would probably only diverge noticeably from the predictions of our best classical theory of gravity (Einstein's theory of general relativity) at the Planck scale, involving both very high energy densities and very short distances and times. The only places where you can get such intense concentrations of energy are under circumstances where general relativity would predict an infinite concentration of mass/energy, i.e. where it predicts singularities. It's under these extreme circumstances that physicists think quantum gravity effects would become significant.
One other subtlety--they actually aren't talking about the singularity at the black hole's center! As Thorne explains, attempts to model realistic rotating black holes by physicist suggest the presence of two other singularities, caused by the fact that waves falling into the black hole (gravitional waves, electromagnetic waves etc.) get infinitely blueshifted (meaning a localized region of infinite--or Planck-scale, in quantum gravity--energy density) well before they reach the center of the black hole. I discussed this in more detail in an answer here, but the basic idea is that if you fall into a black hole, depending on your trajectory you can pass through either an "infalling singularity" caused by matter/energy/gravitational waves that fell in after you, or an "outflying singularity" (discovered as a theoretical possibility only recently, in this 2012 paper) caused by matter/energy/gravitational waves that fell in before you, but where "a tiny fraction of that stuff is scattered back upwards towards you, scattered by the hole's warpage of space and time ... The upscattered stuff gets compressed, by the black hole's extreme slowing of time, into a thin layer rather like a sonic boom". Both these singularities may be "gentle" singularities in the sense that although the tidal forces go to infinity as you pass through them, the tidal forces only become really large for such a brief period of time that they may not rip you apart (as they would if they lasted for longer, see spaghettification). And in chapter 28 Thorne explains the idea is that Cooper did manage to pass through the outflying singularity before being scooped out of the black hole by the tesseract, and thus TARS was able to obtain the needed information about quantum gravity.
So that's the long answer to your question—probing the singularity would allow them to discover the correct theory of quantum gravity, which would allow them to find the correct form of the Professor's nonquantum equation after discarding the quantum fluctuations, which would allow them to understand the anomalies and control the local value of the gravitational constant G.
