A lot of answers here seem to be answering a the related but un-asked question of "How does the lucky person get away with it or maximize winnings?"
The answer to the actual question is rather simple and short...
How can the casino prove I'm cheating?
If you are not cheating, they cannot prove that you are. How would they? The casino cannot prove something when there is no proof.
The casino could try to plant fake evidence. This is unlikely to happen unless you are breaking the bank for any given, specific casino. Some people do walk away with millions, and the casinos don't plant evidence against them all, so why would you be different?
If you are raking in Billions from 1 single casino, then maybe they would try something. At that point, though, what they are doing is illegal, so just use your imagination. Maybe they rig their own roulette game in a fashion that you could for cheating and they say you did it, maybe get false witnesses. There are 1000 things they could do here.
Speaking of illegal activity, casinos are highly regulated in a lot of ways. It is illegal even for the casino to tamper with their own devices to cheat in the casino's favor, just like it is illegal for you to tamper with them to cheat.
That said, they will definitely suspect you of cheating. There have been times when people have cheated, the casinos have looked into them and proven they did cheat, and those people have gone to jail.
What Can (or Will) the Casino do?
Even though the casino cannot prove you are cheating, that doesn't matter. They don't need to prove it. They can and will simply kick you out. It is that simple.
If you keep winning too much money in a way that does no good for the casino (provides no positive publicity for casino, causes nobody around you to bet more, etc.), eventually someone will come over and just force you to leave.
What Else is the Casino Likely to do?
Planting evidence is unlikely, as mentioned above, since that could get the casino in big trouble. However, that does not mean the casino will just roll over. I will also go into the situation where you try to get one last big win so the casino has little time to react, but I will cover it from the casino's point of view, per your question.
I have heard of people being denied their winnings before. Even if this is illegal (some have suggested it is, and I would agree), that does not bother the casino if they do not lose out. This, then, becomes just another bet, another game of chances. Odds are that people who are not given all the chips they deserve are not going to take legal action against the casino, and even if some do then the casino just pays you what you were owed in the first place. That is: there is no negative for the casino since they lose (approximately) the same amount whether they pay you now or pay you when your lawyers threaten them and so there is little incentive to bully you into leaving without all earnings, but there is much positive for the casino if you don't send lawyers. So it's a "win/not-lose" situation for the casino to bully you into not taking that last big winning when they are kicking you out.
Further, if you have no witnesses or evidence, it may be hard to prove that you were denied what was rightfully yours. If you leave without all your winnings when the casino bullies you, what proof do you have that you actually were denied what was yours? If you were keeping a low profile, you likely left few or no witnesses. This makes it even easier for the casino to withhold what is yours.
Summary
The casino can't prove anything but doesn't need to. The casino can and will kick you out. It is possible they might try to bully you into leaving without all your money.
