Pleaaaaase tell me the answer to the puzzle? I didn't quite get it... They EITHER gets 1 or 0??
No, they get a random number from zero to one (inclusive) but no number in that range is preferred to any other, that's what "uniform" means. For example, you might get 0.5 or 0.34598231 or 0.99991413872 etc. But of course the likelihood is zero of getting a number with finite precision like that because such numbers are so few compared to all the real numbers in the range.
One way to think of it is as a limit of a sequence of finite sets of numbers. For example, a finite instance would be to get a random decimal fraction between zero and one with 100 or 200 or 1000000 digits of precision and then let the amount of precision grow very large.
If you wanted to simulate the game very closely in a program, you'd use a random number generator, let's say one that produces integers from zero to 232-1 and then divide by 232 to obtain a fraction. Both players get one and they can keep it or turn it in a draw another random number the same way. The numbers they end up with are compared and whoever has the greater one wins.
And so the puzzle is to devise a strategy a player can follow to ensure he wins with probablility at least 1/2 no matter what strategy the other player follows.