You could try to apply an oil stain across the entire bottom of the drawer. Perhaps boiled linseed oil? I would try that first. If that still doesn't cover up the stain, remove the drawer bottom from the drawer and lightly sand with a random orbital sander. Be sure to sand everywhere, not just the stain. Sand at a fine grit, like 200 or above. That's probably a wood-veneered drawer bottom, and there's probably only a very thin layer of hardwood veneer on top of inner layers of worse looking wood.

If sanding proves impossible, replace the drawer bottom. Make a new drawer bottom the same size, shape, and thickness out of pre-sanded hardwood veneer plywood. Select a plywood of the same thickness as the drawer bottom. Cut it to size on a table saw, a track saw, or a circular saw with an edge guide. Remove the old drawer bottom and slide the new one in.

Or, if you really want an easy fix, don't even try and fix or replace the old drawer bottom. Get a thin piece of hardwood veneer. Cut that to the size of the exposed portions of the drawer bottom. Then simply glue the veneer on top of the old drawer bottom using wood glue. Instead of clamps, the new veneer can simply be held down with a heavy weight placed atop it. You lose a tiny amount of the drawer's volume this way, but you don't have to disassemble and reassemble the drawer. You could also just line all the drawers with felt. This would cover up the stain while not looking out of place.

In the future, prevent this from happening again by treating the drawer bottom like any other piece of wooden furniture. Would you set a sweating drink on top of a wooden table without a coaster? Well the same applies to your toys. Keep them in a non-permeable container or ensure they are thoroughly clean prior to placing them on a wooden surface.