How exactly do they stretch they tires so they stay? Just pull and it eventually catches or something?
You use brake fluid and spray around inside the tire on the rim while the tire is already mounted, then light it while it has weights on it and the beads should seat right away, then start filling it with air before the vacuum inside the tire sucks itself in.
http://www.youtube.com/watch?v=n-LnkEUAImU
Another method is to use a bicycle tire tube, explained in the video below.
http://www.youtube.com/watch?v=lzxHYEdA8Qo
Or use a ratcheting tie down straps and tighten the s**t out of the circumference of the tire forcing the beads onto the rim.
Here's a good example of cramming in wide and larger rims with thinner tires to fit on a lowered car.
In performance terms, it is popular among drifting culture, because the tire has less contact patch, and it is easier for it to slide sideways because the edges of the tires/sidewalls are sloped. Stretched tires are popular among Euro cars and old school imports like the AE86.