In the episode "The Late Philip J. Fry," Fry and friends go billions of years into the future and witness the end of the universe, with all the stars and galaxies exploding and fading into nothing. This roughly jibes with one theory of the end of the universe, in which all matter and energy becomes so uniformly spread out it no longer interacts with itself, creating a stasis known as "heat death." Professor Farnsworth's throwaway line, "There's the last proton decaying," is a little iffy -- in the most common physics models, protons don't decay.
The end of the universe isn't the end for our characters, however. They witness a new Big Bang and the birth of a new universe, which plays out identically to the old universe (even down to Leela waiting for a chronically late Frey in the same restaurant that she did in the old universe). While the concept of the Big Bang is accurate in a very general sense, "Futurama" doesn't depict it accurately. The Big Bang was not an explosion in space; it was an explosion of space. In the Big Bang, space itself expanded from an infinitely small point. You couldn't witness the Big Bang from an external location unless you were outside the universe (and because they eventually return home we know that Frey, Farnsworth and Bender are still inside the universe).
The repeated death and rebirth of the universe is conceptually accurate in some cosmological models, although it could happen by many different mechanisms. Most commonly, the universe could shrink back down to a point instead of experiencing heat death. That point (a singularity) would eventually undergo another Big Bang and restart the process.