I have heard about this type of thing happening have never had to deal with it myself.
You can choose to turn off the house water but do not need to. As there are screws behind the plate that act as valves for the incoming lines for this handle.
First remove the handle with an allen/hex wrench and unscrew this black piece (center screw) pay attention to its positioning.
Under the black piece are two white pieces stuck together, one has a protruding tab that is pointing to about 1:00 on a clock and the other (behind it) is pointing straight up. Remove them, remembering what was pointing where. Remove the metal sleeve
This is what the two white pieces look like
You will now see what is called the cartridge, this is what has failed. Directly to the left and right of the cartridge are the two screw valves that will stop the water.
First remove the retaining clip with a pair of pliers
Then insert a flat head screwdriver next to the white piece you see here and pry it outward, once you get it moving then pull it the rest of the way out with pliers. No water should come out of here when its removed if the valves are turned off.
This is the cartridge, $40, reverse the steps and you'll be up and running in about 15 minutes.