The big crunch theory is one possible scenario for the end of the universe, but what makes it more interesting, is that it is one possible scenario for the beginning of this universe.
The theory almost predates the discovery that the universe was expanding from a big bang, and makes a lot of sense based on "simple" ideas like gravity.
The reason it has been largely dismissed by science over the years is observations that the universe is expanding faster, and theories that suggest that gravity cannot slow the expansion, even over an infinite amount of time.
First, let's look at why the universe is expanding faster. Nobody knows. They think it's something called "dark energy", and what that means, is they can't see it, but they can assume that something is there, based on observations of the effect it seems to have on other objects.
In order for the expansion of the universe to be accelerating, something must be pushing it, because an object needs propulsion just to accelerate in empty space, let alone away from the gravity of the inner universe.
It's a possibility that this "dark energy" is not dark at all, it's light. Light is hitting our galaxy all the time, from stars burning in the inner universe, going back billions of years.
Light particles hit us all the time, and give a little nudge to the galaxy, causing propulsion. Without propulsion, the universe could only slow in expansion, and start to slow to a stop, then fall back into gravity.
Gravity has no limits. It's a weak force, but the gravity of our nearest neighbouring galaxy Andromeda, will mean that it will smash into us within a few billion years.
Dark matter is the name given to all the matter floating around that we can't see. It makes up perhaps as much as 99.9% of all the mass in the universe.
Dark matter is probably black holes by definition, because when matter gets cold enough, it collapses. At the point of absolute zero, the electron stops spinning around the nucleus, which takes away 99.999% of the volume, (approximately), but not the mass.
The Big Crunch Theory
Let's take a look at what gravity is. Gravity is the bending of space/time. It bends not molecules, not dust, but actual space, in a way we can't really comprehend.
Everything has gravity, from the largest black holes, to the tiniest sub-atomic particles, because they bend space/time just a little bit as well.
There is a theory that black holes will eventually evaporate, but if you look at the way gravity works, there is no way anything could ever leave the event horizon of a black hole by just evaporating off it, as those particles would be subject to the same intense gravity.
As I said before, gravity has no limits, and light does. Light will eventually stop being emitted by stars, because they will all burn out, and turn into black holes, which suck in light.
This may sound like a simplistic explanation, but the fastest the universe could ever expand is the speed of light, and of course there are light particles traveling off at that speed into the void.
That's the fastest anything can travel. Gravity has no limits, so what happens? It gradually slows these particles and objects down. To do the calculation would require a degree, (which I don't have), but I can explain the equation.
Gravity slows down the light by a certain amount, then the light is further away, so gravity slows it down by a smaller amount, or it takes longer to slow it down by the same amount.
There is an infinite reach of gravity, and not an infinite amount of speed of those objects, so eventually, everything will be sucked back in by gravity, even if it takes an almost infinite amount of time.
Nothing escapes gravity, ever. That means the universe is eternal, moving in cycles of big bang, big crunch, big bang forever. This is supported by the first law of thermodynamics, (energy cannot be created or destroyed, it only changes form), the second law of thermodynamics, (needs a clause at the end, saying things will fall apart, slow down, gain entropy, and get sucked in by gravity, until the end of this universe, and the start of the next), and it's supported by causality, (every action has an equal and opposite reaction).
Energy equals mass multiplied by the speed of light squared, and mass equals energy divided by the speed of light squared. The law of the conservation of mass and energy says that in an enclosed space there is always the same amount of energy and mass, and the law of gravity says that what goes up, must come down, (or that gravity has no limits, and there are limits on the speed and acceleration of mass and energy away from gravity).
As for complicated equations based on mathematics, I don't understand them. I don't undertand branes, and multiple dimensions. I know for a fact that most of these theories based on equations can be proven wrong, simply by the fact, they can't all be right, therefore nothing is proven by mathematics.
To do an observable experiment that showed whether or not the big crunch does happen at the end of each universe would be absolutely impossible, like trying to observe the big bang, therefore, the big crunch theory can never be proven, or proven wrong.