How does he comprehend a fourth and fifth dimension so readily when his brain has been stressed as far as it has? How does love, something you cant quantify in any real way, give him the power to navigate the very fabric of time and how can we begin to accept that his communications held up when his own ship stopped working?
The tesseract is a fifth-dimensional space but it's deliberately constructed as a three-dimensional structure for Cooper so that he can understand it. And it's not as simple as "the power of love transcends dimensions" -- that's kind of what it boils down to, but the mechanics of that scene are that Coop is able to distinguish one moment from the next in his daughter's life because he knows and loves her and has a concept of her personal timeline. The 5D beings see the past, present, and future as one amorphous stream of events, so they can't really tell apart one moment from another.
How does he get out of the center of a black hole, one tiny body at the end of life support, right in front of a ship that happens to be passing by. Providence? That just leaves a really gross non-sciency taste in my mouth, for a movie that seems to really like science.
Providence, as in the work of far-future hyper-evolved humans.