In North America, women keep bills and coins in their purses next to used tissues, and studies have shown that men's wallets have an even higher concentration of bacteria than women's purses. We sit on and, on hot days, sweat on our money all day. Let's face it, the stuff is pretty disgusting. However, if somebody hands you a twenty, your first reaction is usually not, "Oh gross, now I need to wash my hands".
In terms of hygiene, it makes sense for waiters to collect currency on trays so they don't have to wash their hands after every bill is settled (assuming it's a different person's job to take the cash off the tray and put it in the till). Could this be the reason the practice started? The connection between disease and microbes would have existed around the time these trays appeared, so it's possible a crusading doctor or someone similar convinced vendors to use these trays.