Interstellar is one of the best films to come out this year. The tragedy is that few people will understand it. Much of the science behind it is theoretically possible, but unfortunately some of it isn't. As someone who enjoys theoretical physics and movies that challenge the mind, I'll try explain it from both sides.
Aside from some great acting by Matthew McConaughey and Mackenzie Foy, the best parts in the movie were journeys through a wormhole and a black hole. The wormhole scene was visually incredible, and I don't feel a particular need to explain the science behind it, as it can be found in many other sources (and the movie itself explained it pretty well). There is also much information about how gravity makes time relative, which is based on Einstein's Theory of Relativity. Time is as much influenced by the curvature of space in relativity as the three dimensions of space are. This is why we collectively call it space-time. The more gravity there is, the more the curvature of space "collapses" time. On a massive planet near a black hole, like the one in the movie, time would slow down immensely because of the localized influence of gravity. Despite the scenes being live-action, everything is really moving super slow compared to everything on Earth (hence the term relativity).
Later in the film, the main character falls into a black hole and floats around inside a tesseract that was created by humans. A tesseract is a four-dimensional cube, where events in time and 3D space become infinite. Essentially, he was seeing space in five dimensions, which is why he could see an infinite varieties of the same thing happening. The idea that a black hole could contain portals to parallel universes and higher dimensions isn't as far-fetched as it seems. Nobody knows for sure what goes on inside a black hole, so it could very well contain a compartmentalization, or matrix, of the entire multiverse. It might be analogous to the central processing unit of a computer, in which an infinite variety of commands can produce different results.
A black hole is a place where gravity is so strong that nothing can escape, not even light. This is why I found it incredulous that a robot could communicate with the main character inside it, but then I remembered that it had recorded quantum data from its surface as their spaceship was swinging around it. Those bits of data, which are called Plank bits in String Theory, are theoretically attached to the bits of data on the interior, via their strings. What didn't sit well with me was the fact that he survived the trip and magically landed outside the wormhole near Saturn. Perhaps in the director's mind the 5th dimension gives one the power to travel anywhere in the universe at will. But if the main character had that power, then why wouldn't he have sent himself back to Earth instead of the middle of space? It makes it seem like he wasn't in control of where he was going, and that his ending up near the portal was purely by chance. It would have been a better ending if he'd just died and continued to communicate with his daughter on the other side.
Director Christopher Nolan has some brilliant ideas, I'll give credit for that. Unfortunately, he seems to do too much with them, making the plots suffer near the end of his films. The same thing happened at the end of The Dark Knight, when Batman, instead of framing the Joker to make Harvey Dent a hero, framed himself. He has succeeded a couple times, with Memento and Inception, and for this reason I feel they are better movies.
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