It does drain into the bilge. Look for a white ribbed drain hose in the bilge. It is pretty long, and may extend toward the bow farther than you can see with the 2 boards out. It is not secured (at least mine was not), so if you see it in the bilge, you should be able to pull it back to get to the open end.
Check for blockage at the end, maybe pour some hot water into the icebox and see how it comes out of the drain hose. You could blow air back through the open end of the hose to see if that clears it.
I never figured out how to get to the bottom of the box where the drain attaches....maybe through the starboard settee storage hatch.
Good luck.
Greg