First, let me transcribe the relevant portion of that podcast starting around 00:24:20:
Well, yet another proposal being circulated that I personally think is promising is the ramjet fusion engine. Instead of using fission power, that is, the power of uranium, which creates meltdowns and all sorts of nuclear waste, why not use fusion, the fusion of hydrogen? Make it a ramjet fusion engine, so it literally breathes the hydrogen inside outer space. Here's how it would work. Think of an ice cream cone, an ice cream cone that moves in a forward direction, scooping up, scooping up interstellar hydrogen as it moves forward. Then it concentrates the hydrogen toward the back of the ice cream cone, heats it up, fuses it so that it creates energy, and then helium gas shoots out the other end. Now, because there's almost a limitless supply of hydrogen gas out there in outer space, 'cause after all that's what the universe is made of, then there's an infinite supply of fuel, and therefore in some sense this rocket will go on forever. Forever! And so you turn it on and it simply continually accelerates until it reaches near the speed of light.
In fact, there's even a science fiction story where the first one of these ramjet fusion engines is built. It accelerates and then it has a malfunction. It's stuck in autopilot so it just keeps on accelerating and you can't stop it. You cant stop it 'cause it scoops up basically an unlimited supply of hydrogen in the forward direction. And as it begins to reach the speed of light Einstein's time distortion starts to come into play. Time slows down inside the rocket ship, so the universe ages naturally but the people inside have stopped ageing, and so from their point of view the ageing of the universe speeds up, and so they witness the formation of stars, the formation of galaxies, the death of stars, the death of galaxies as they rocket out of control going faster and faster and faster getting closer and closer to the speed of light; and in the final end of this novel these astronauts witness the creation of another universe because they first of all witness the death of our universe. In the novel the universe begins to contract, temperatures begin to rise, and they are able to go right through the Big Crunch and witness the birth of the next universe. Well of course that's pretty far-fetched, but at least in principle it does seem to obey some of the laws of physics.
There is no mention of a black hole here, just a starship with a ramjet fusion engine which has lost its brakes, so that the astronauts fly on and on, witnessing the end of our universe and the birth of a new one. It's clear that the speaker is talking about Poul Anderson's 1970 novel Tau Zero, which was also the answer to this old question. (Tau Zero was an expanded version of Anderson's 1967 novella "To Outlive Eternity", but the podcast refers to the novel, which of course is much more famous.)
Here is a synopsis from the Internet Speculative Fiction Database:
During an epic voyage to a planet 30 light years away, the engines of the starship the 'Leonora Christine' are damaged. Unable to slow down, it attains light speed (the tau zero of the title). The disparity between time for those on board and external time becomes impossibly great. Eons and galaxies hurtle by, and the crew speed helplessly into the great unknown.
Tau Zero follows the crew of the starship Leonora Christine, a colonization vessel crewed by 25 men and 25 women aiming to reach a distant star system. The ship is powered by a Bussard ramjet, which was proposed 10 years before Anderson wrote the book. This engine is not capable of faster-than-light travel, and so the voyage is subject to relativity and time dilation: the crew will spend 5 years on board, but 33 years will pass on the Earth before they arrive at their destination. The ship accelerates at a modest constant rate for most of the first half of the journey, eventually achieving an appreciable percentage of the speed of light, and the goal is to decelerate at the same rate during the second half of the journey by rotating the Leonora Christine 180° about its trajectory axis and continuing to fire the Bussard ramjet. However, the Leonora Christine passes through a nebula before the half-way point, damaging the Bussard ram scoops which were to have been the deceleration module. Since the Bussard engines must be kept running to provide particle/radiation shielding, and because of the hard radiation produced by the engines, the crew can neither repair the damage nor turn off their ramjet.
[. . . .]
As there is no hope of completing the original mission, the crew increase acceleration even more; they need to leave the Milky Way altogether in order to reach a region where the local gas density, and the concomitant radiation hazard, are low enough that they can repair the decelerator. The ship's ever-increasing velocity brings the time dilation to extreme levels and takes the crew further and further away from any possibility of contact with humanity. The initial plan is to locate and land on a suitable planet in another galaxy. Millions of years would have passed since their departure, and in any case they would be millions of light years from Earth. However, they find the vacuum of intergalactic space insufficient for safety; they must instead travel to a region between superclusters of galaxies to make repairs. They do, but the extremely thinly spread matter is then too dispersed to use for deceleration. They must wait, flying free but essentially without the ability to change course, until they randomly encounter enough galactic matter to try to decelerate enough to search for habitable planets. To make the waiting time shorter, they continue accelerating through the first several galaxies they encounter, more and more closely approaching the speed of light with tau, or proper time, decreasing closer and closer to zero.
[. . . .]
The storyline is similar to that of the long poem and later opera Aniara, in which the ship was unable to stop and doomed to travel endlessly, but Tau Zero has a more upbeat ending (albeit one that does not conform to modern thinking on the evolution of the universe). By the time the ship is repaired, tau has decreased to less than a billionth and the crew experience "billion-year cycles which passed as moments". But by the time that they are ready to attempt to find a future home, they realize that the universe is approaching a big crunch. The universe collapses (a process the starship survives because there is still enough uncondensed hydrogen for maneuvering, outside the monobloc) and then explodes in a new big bang. The voyagers then decelerate, examining potential star systems. They eventually disembark at a planet with a habitat suitably similar to Earth, on which the vegetation has a vivid bluish-green color.