Why do the two bottom wires on my GFCI have power?
Your top wires are probably your LINE wires. Basically the wires that come from your panel. Your bottom wires sound like your LOAD wires. The way a GFCI works is you give the plug LINE power to protect that plug. But if you want to protect more plugs than just that one plug, you put the other wires on the LOAD of the plug. Basically anything down stream is protected if it is fed from the load side of the plug. So a regular looking non GFCI receptacle could stop working one day, because it tripped the GFCI that feeds it. So you would reset the GFCI and the plugs would work again, Your dishwasher, since it does not want to be GFCI protected (would probably trip the GFCI) is attached to the line wires. So it is not GFCI protected. It gets power from before it even goes onto the plug.
When installing your plug, look at the back of it. The top two terminals are usually labelled LINE and the bottom is labelled LOAD. Don't forget the silver screw is your white wire and the copper screw is your black wire. Green is ground.