Good question. The electrons do reach the anode and collect there but only temporarily. Remember that the anode is connected to the cathode by a battery. So the battery forces the excess electrons back to the cathode through a wire. This is just a current in the wire.
No problem. If the anode is not connected by a wire then it will charge up to a negative potential and eventually repel any more electrons from collecting.
LeeH
(published on 11/13/2007)