I've found it best to cut a circle about 1" smaller diameter than the mirror out of 3/8" or 1/2" plywood, pad this with 2-3 layers of towel and press against the mirror in order to compress the 3 springs behind the mirror enough to remove the snap ring holding the mirror in place. Holding the mirror in slightly, you can remove the ring and the mirror will come right out. Installation is the reverse of these steps.
The reason I use the plywood is that it spreads out any pressure against the mirror to the full face of the mirror. I've seen mirrors broken (snap in the center) when pressing a finger against the mirror to compress the springs. Not saying it can't be done, I just don't choose to risk breakage of the old or new mirror unnecessarily. Just my way of doing it!!