Comprehension help required
Solution 1:
There are three common uses for this phrase but all have a similar meanings. The extended example you post doesn't really make the phrase clear through context so it isn't much help.
The first usage is one where someone gives another person money to use as a loan or gift and then wins the money back later in a contest. While they did win the money, it doesn't really feel like winning because they only walk away with what they already had when they came in. This is the most likely usage to match the example you gave.
Alice: Can I borrow $5? I want to play some cards with you.
Bob: Sure... but then I'll just be winning my own money. What's in it for me?
A second usage is when you win back money that you lost in a previous game. A classic example of this is the final scene in the poker movie Rounders. The hero is playing a game against The Russian who won thousands of dollars from him early in the movie. During this scene, the hero beats The Russian and is happy with his winnings. The Russian goads him into playing another game by saying, "After all, I am paying you with your money."
The third usage is when you steal someone's money and then use it to gamble against them. A classic example of this is when someone steals their enemy's wallet and uses the cash found in the wallet to enter a poker game. If they "lose" their enemy will simply be "winning their own money"; the point of this trick is that they can gamble without much risk. Even if they lose, they aren't losing their own money.