In a word: mutation.
If you step back and think about it, multicellular organisms are really, really complex systems. Different cells have to do different kinds of things and interact with each other, they form organized tissues and organs and so on. That much is obvious.
For all of this to work in concert, cells have to grow and divide (to replace dying cells) in a nice orderly fashion/at a nice orderly pace. Since that's a complicated process, lots of the genes in any given cell play a role in managing how long that cell lives/how fast it divides (i.e., cell cycle regulation).
So now, if DNA in a given cell gets mutated, there's a chance that mutation will hit one or more of these genes that are involved in cell cycle regulation. And you can imagine there are two extremes of what might happen: On one extreme, the mutation could mess up the cell cycle in such a way that it's broken (the cell can't grow/divide, things get screwed up, and it dies off); on the other extreme, the mutation could instead destabilize the cell cycle in such a way that it's "kicked into overdrive."
In the first example, we wouldn't observe the effect; the mutated cell would just die off, no big deal. But in the second example, with the cell cycle in overdrive, that cell would barrel ahead, growing fast and dividing faster than the cells around it. And since each of its child cells would have the same DNA and thus have the same "kicked into overdrive" mutation, they'd divide too quickly as well, and so would their children…and eventually you'd have a crazy mass of out-of-control cells. That's a tumor.
p53 is a tumor suppressor because it effectively "sounds the alarm" when it notices DNA damage (mutation)—meaning, when it finds DNA damage, it initiates several processes to try and control/fix/stop it. This is important because if that DNA damage is like the second example above, it could lead to a cancer that harms or kills the whole organism.
If the p53 gene itself is mutated, then the cell has lost one of its major defenses against damaged DNA, and thus there's a much greater chance that a cancer-causing mutation in some other gene could go unchecked. That's why damaged p53 is found in lots of cancers: because the cell lost a big defense mechanism against cancer, and then cancer happened.