Paintball is indeed used in military training. In terms of feel and handling, a paintball gun has little in common with an actual firearm, but paintball is still good practice for combat situations. Mainly, it lets soldiers and police practice strategy, camouflage and the fine art of not getting shot. However, while paintball has become a part of military and police training, that's not where it originated.
The first guns were developed in the 1970s for use in forestry and agriculture. Foresters used the guns to mark certain trees (for research, planning trails). The guns were also used by farmers to mark cattle.
At some point, it occurred to a few foresters or farmers to shoot the guns at each other, and the game of paintball was born. But things didn't really get going until 1981, when a group of 12 weekend warriors got ahold of some forester guns and used them to play a grown-up version of "capture the flag."
In this game, which is still the predominant paintball activity, two teams try to find and steal the other's flag while protecting their own flag from capture. Players are "out" of the game when they get hit with a paintball, and the referee decrees that they are down. Referees are also there to make sure nobody makes physical contact with another player: This is one of the most important rules. A paintball game typically lasts from 15 to 40 minutes, but players may play for six hours or more at a stretch. You can hold a paintball game with only a few people on each team, or with hundreds of people on each team. To find out about other variations on the game of paintball, check out this site.
The original 12 paintball enthusiasts had a lot to do with launching the sport. Soon after their first game, they bought up hundreds of tree-marking guns from the manufacturer (a company called Nelson) and began selling them to the general public. The idea caught on pretty quickly, and in 1982, the first paintball field opened in Rochester, New York. There are now paintball fields, as well as indoor paintball arenas, all over the world.
Here are some interesting links: