You can see my answer here for a lot of details on the physics problem they were trying to solve, but the short answer is that they had found gravitational "anomalies" suggesting that the gravitational constant G actually wasn't completely constant after all, and if they could learn to control it, they could find a way to leave the dying planet in massive numbers by temporarily making gravity much weaker (so the energy needed to achieve escape velocity would be much smaller). But to understand how to control it, they needed a theory of quantum gravity, and to find the right theory they needed measurements that could only be made near the singularity of a black hole (again, my answer above goes into more detail). So Cooper was giving Murph information from TARS' analysis of measurements they made inside the black hole, before they were rescued by the tesseract (which had been created by advanced beings, possibly descendants of humanity, in a higher spatial dimension).
In The Science of Interstellar by physicist Kip Thorne, chapter 31 talks about them using quantum gravity to decrease the Earth's gravity and get the huge colonies off the Earth:
Early in Interstellar, when Cooper first visits the NASA facility, he
is shown a giant, cylindrical enclosure being constructed to carry
thousands of humans into space and house them for many generations: a
space colony. And he's told there are others being constructed
"How does it get off Earth?" Cooper asks the Professor. "Those first
gravitational anomalies changed everything," the Professor replies.
"Suddenly we knew that harnessing gravity was real. So I started
working on the theory—and we started building this station."
How did it get lifted into space? The key, of course, was the quantum
data (in my scientist's interpretation, the quantum gravity laws)
that TARS extracted from Gargantua's singularity (Chapters 26 and 28)
and Cooper transmitted to Murph (Chapter 30).
Murph must have figured out how to reduce Newton's gravitational
constant G inside the Earth ... In my interpretation, with Newton's G
reduced inside the Earth to, say, a thousandth of its normal value
for, say, an hour, rocket engines could lift the enormous colonies
As a byproduct, in my interpretation the Earth's core—no longer
compressed by the enormous weight of the planet above—must have sprung
outward, pushing the Earth's surface upward. Gigantic earthquakes and
tsunamis must have followed, wreaking havoc on Earth as the colonies
soared into space, a terrible price for the Earth to pay on top of its
blight-driven catastrophe. When Newton's G was restored to its normal
strength, the Earth must have shrunk back to its normal size, wreaking
more earthquake and tsunami havoc.
But humanity was saved. And Cooper and ninety-four-year-old Murph were
reunited. Then Cooper set out in search of Amelia Brand in the far
reaches of the universe.