Been awhile since I've done this ...and honest, I meant no disrespect above, but it's an interesting and related procedure on the "real thing" I thought might be an interesting parallel....
That said, it's been awhile since I did one of these, but if I recall, it works like this:
1.) Removing the old tire is easy ... a screwdriver or knife blade can be worked under the edge of the tire, prying it up and ulitimately off the wheel. This will distort and or break the tire, but it's worn out and has no "second life" anyway, so this doesn't matter ... just don't be so forceful that you bend or gouge the wheel trying to get the tool underneath.
2.) The new tire can then be slipped on with a little care, by starting in one point and then working outwards with the thumbs around the rim of the wheel until the whole thing is popped into place.... it has a little stretch in it, enough to slip over the wheel like a very tight rubber band.