You can use a screwdriver to remove the old rubber catch, but you run the risk of scratching your porcelain or stainless steel sink. Instead, after you unplug the disposal, plunge the head of a serving spoon past the rubber splash guard. Then, pull it straight up -- the splash guard should snap off with ease.
Buy a tube of buytl caulk at any hardware or plumbing supply store. Apply a thin ring of caulk to the outer rim of the new rubber catch and slide it in place. This will adhere the catch to the drain and prevent it from popping off or losing its shape.
Rather than bother with removing the old rubber catch, you can simply snap down a new one over it. This may seem rather slothful, but there are benefits to having two rubber catches in place. Two catches make for a quieter disposal and far less splash-back to contend with.