What is happening here is fairly simple: Since each state in the system occurs with the same probability, the distribution that arises is the one that maximizes the entropy. Since the total amount of money is conserved, this means that the mean of the distribution is fixed. The maximum entropy distribution in this case has a geometric shape, which is skewed:
Yep, it's just a simple counting argument, you see the distribution that has exponentially more states than any other, subject to constraints. Specifically, you are allowing the range [0,∞). If you had constrained it to a fixed range of money (e.g. there is corrective feedback when you get too much money), then you would have seen a more uniform distribution.
https://en.wikipedia.org/wiki/Maximum_entropy_probability_di...