How did Cooper avoid spaghettification when entering the black hole at the end of Interstellar?
This is adressed in the film shortly, if only a little superficially. When Cooper is about to leave Mann's planet homeward bound, Romilly asks him to make a trip along the black hole. And while the idea of Cooper going into it rather than just TARS alone isn't yet prevalent, he does adress the difficulties of entering it:
Romilly: Gargantua's an older spinning black hole. It's what we call a gentle singularity.
Romilly: They're hardly gentle. But the tidal gravity is so quick that something crossing the horizon fast might survive.
The point here is that not every black hole causes spaghettification when crossing its event horizon. And in fact Gargantua is a humongous black hole (with about 100 million solar masses), or more scientifically speaking, a supermassive one, for which crossing the horizon without experiencing spaghettification could be possible, according to Wikipedia:
The point at which tidal forces destroy an object or kill a person will depend on the black hole's size. For a supermassive black hole, such as those found at a galaxy's center, this point lies within the event horizon, so an astronaut may cross the event horizon without noticing any squashing and pulling, although it remains only a matter of time, as once inside an event horizon, falling towards the center is inevitable.
And this is part of what Romilly alludes to in the movie when calling it a "gentle singularity" (although, there's a little more to it adressed in other answers).
This relation is also alluded to a little by the film's executive producer and scientific advisor Kip Thorne in his book The Science of Interstellar, albeit in the context of Miller's planet rather than Cooper's entering of the horizon:
At so close a distance, Gargantua's tidal grivational forces [...] are especially strong. They stretch Miller's planet toward and away from Gargantua and squeeze the planet's sides [...] The strength of this stretch and squeeze is inversely propoportional to the square of Gargantua's mass. Why? The greater Gargantua's mass, the greater its circumference, and therefore the more similar Gargantua's gravitational forces are on the various parts of the planet, which results in weaker tidal forces. [...] Working through the details, I conclude that Gargantua's mass must be at least 100 million times bigger than the Sun's mass. If Gargantua were less massive than that, it would tear Miller's planet apart.
The same principle applies to the spaghettification Cooper experiences when entering the black hole, since that is based on the strength of the tidal forces, too, i.e. the conflict between simultaneous stretching in one direction and squeezing in the other.
Note though, that while Cooper and TARS can basically survive the black hole's event horizon without getting literally torn apart, the tidal forces get much stronger when approaching the black hole's center, along with a ton of other problems of entering a black hole. But by that point they were already saved by the Tesseract inside of Gargantua, as explained in the answers to this question (and various others linked from there, I suppose).