When you get freshly printed money, usually at a bank, the pieces of paper are dry, perfectly flag, and they actually are crisp--they make a slight rubbing noise as they slide against each other and will make a little noise if you crumple them. They have a slight tendency to stick to each other. Our local bank has a counting machine at each teller's window now, but not so long ago, a teller who was handed new money would put little rubber thimbles on their finger, and kind of deliberately give each bill a little twist or tug as they counted it to make it less sticky.
There is a tradition, half-true, half-symbolic, that money is "dirty." "Crisp new bills" give you the idea that nobody has ever touched them before, so they are nicer and "cleaner" than old bills, even if they don't actually buy anything more.
Actually if I give e.g. grandchildren gifts of money I'll go the bank and get crisp new bills to give them.