I'm watching MIT OpenCourseWare lectures on general relativity and not too long into the first lecture the professor stated that the Kerr Black Hole solution allowed for travel between universes. How is that possible to know? How does one derive this/come to this conclusion?
It is correct that the Kerr black hole solution of GTR allows travel between universes. However, that does not mean that if you actually jump into any kind of black hole that you could go to another universe.
To motivate the resolution to this conundrum, let's start off very easy: suppose you stand on the ground with a ball in your hand, and you throw it with some initial velocity. For simplicity, let's ignore everything except a uniform gravity. Mathematics will then tell you that the ball follows a parabolic arc, and when and where the ball will hit the ground. And if you take the resulting equations too literally, then it will also tell you that the ball hits the ground twice: once in the future, once in the past. But you know the past solution isn't right: you held the ball; it didn't actually continue its parabolic arc into the past.
A morally similar kind of thing occurs for, say, a Schwarzschild black hole. If you look at it in the usual Schwarzschild coordinates, there's a problem at the horizon. Mathematics will then tell you that the problem is just with the coordinate chart, and that there's actually an interior region to the black hole that becomes apparent in different coordinates. And if you do this generally enough, it will tell you that there's more to it than even that: there's also white hole with a reverse horizon and its exterior region--another universe. This full "maximally extended" Schwarzschild spacetime has this other universe connect to ours via an "Einstein-Rosen bridge" and then "pinch off", producing separate black and white holes.
Of course, that too is an artifact of mathematical idealization: and actual black hole is not infinitely extended in the past and future; it was actually produced by something, a stellar collapse. (And the "bridge" isn't traversable anyway; one will be destroyed in the singularity if one tries.)
Finally, on to the Kerr solution, it's a bit better because formally the singularity is avoidable, unlike the Schwarzschild case. However, it's still physically unreasonable: in addition to the fact that actual black holes aren't eternal, the interior of the Kerr solution is unstable in regard to any infalling matter, which will perturb the solution into something else entirely. Therefore, it cannot be taken as a physically meaningful. Still, it is true that full Kerr spacetime contains a way into another universe--in fact, infinitely many of them, chained one after another.
If you're interested in the details of its structure, you could look at some Penrose diagrams of those black hole solutions.
"Allowed for" does not mean "necessarily cause".
What the professor implied is that the solutions look, from a mathematical p.o.v., just like what you would expect from a bridge between universes - IF multiple universes exist, and IF the bridge is passable.
That's all there is to it. A mathematical solution that looks like a bridge. But has it ever been verified experimentally? No. Do we have proof that other universes exist? No.
We have the math that describes what looks, for all intents and purposes, like a door. But is the door separating this room and another room, or is it just a fake door built into a solid brick wall, like in movie comedies? We don't know. Would the door open at all? We don't know. Has anyone actually seen such a door yet? No.
That doesn't mean the professor was wrong. It only means this is just a hypothesis at this point. We don't know yet if reality matches it or not.