Why do older players keep their money under the edge of the board?
This has most likely been passed down throughout the generations as "the way to do it", without any actual backing reasons, but there are some situations where it is beneficial to put the money under the edge of the game board.


Small playing areas (such as a small table) where space is at a minimum, so it makes sense to conserve space by putting money under the edge of the board.
Playing outside (where it can be windy) or inside with a fan can cause the money to be shifted constantly, so sliding it under the board allows it to be pinned down.
In games where money is not the driving factor (such as Monopoly, where the Title Deeds are more important), it helps to keep your focus on what matters.
It makes it easier to keep the denominations in separate piles (if that applies) as you don't need to worry about sifting through a single pile to find the right denomination.


Joey has already mentioned the playful/imagination aspect of it which may also apply in some cases.