undo some screws... depending on the year of your civic, there will probably be one in your door handle, one in the arm rest, and one or two by the speaker.
then just take a prybar, or a long screwdriver and go behind the plastic, and infront of the metal... careful not to scratch of course

best to work from the bottom up on removing the panel.
chances are you might break a door retention clip or two, soo pick up some just incase.
then once its removed, unplug the speaker, and remove it (same with any other electrical connection), and probably best to use a steam cleaner on the panel.
then febreeze the inside of the car.