Stolen bases are as old as baseball and the underlying rules are pretty firm. Although a runner has the right to stand on an unoccupied base, where he is safe from being put out, he is not required to do so and may work his way toward the next base or "lead off" in order to get an advantage when the ball is hit. But there is no rule requiring him to wait: he can even run right to it if he wishes regardless of what the pitcher and batter are doing. To do so, of course, he must enter the danger zone between the bases where he can be put out by a fielder with the ball. This is especially dangerous between pitches, when the pitcher is standing on the mound with the ball and can easily throw it to an infielder guarding the base. A thrown ball is faster than a running player, after all.

The precise definition of a stolen base for scoring and statistical purposes has changed over time, which helps us understand the concept's roots. Initially, any time a runner advanced because of running speed and skill, it was considered a stolen base. This included extra bases during a play initiated by a hit. The expectation was that a hit would only allow the runners to advance one base, so a runner who got two or more bases had stolen the extra bases. This definition was used for scoring and statistics from 1870 to 1898 and helps explain how Hugh Nicol's record number of bases stolen in one season (138) still stands: this was from the 1887 season and many of his steals wouldn't be scored as such today. The definition of a stolen base has subsequently been revised so that only an opposed effort to take a base unaided by a hit counts. (If the fielding team just lets the runner take a base, this is scored as defensive indifference. This does happen when the fielding team is winning by a lot.)

But look at what still remains: a stolen base is the product of the runner's speed and skill, now fully separated from the hitter's contribution.

I would say the most important rule contributing to the stolen base is the balk rule, which basically says that if a pitcher on the mound and appearing ready to pitch begins to pitch or throw the ball to a base, he must carry it out. The pitcher is very limited in his ability to "fake out" the batter or runners. The penalty for a balk is that the runners may advance. Although it has been refined over the years, a balk rule hitting basically these points has been present since 1898. In 1950, it was specified that the pitcher had to stand in a ready position for at least one second before pitching. Once the aggressive runner sees that the pitcher has taken position, he may be tempted to take his chance. Indeed, this enforced pause led to an immediate uptick in steals during the 1950 season: stealing had been pretty unpopular during the 1930s-40s.

You mentioned the possibility that stealing was added to the game. As a matter of the history of baseball, this is not quite correct, but it does describe the experience of a kid who grows up playing baseball. When young children play, stealing or leading off is generally not allowed at all, and in Little League, runners must remain on their bases until the pitch reaches the hitter. This is because basic fielding skills are often either lacking or very uneven, giving the runner a big advantage. Think of a game being played by eight-year-olds: if the runner on first just starts hustling toward second, the fielders also being eight-year-olds are frankly unlikely to successfully put him out, and so stealing would dominate the game. The full stealing and balk rules are typically introduced in high school, by which point fielding skills have developed.

The way baseball is taught, then, may lead to the misconception that stealing bases was added to the game. It was not; it is inherent in how runners interact with bases, which is truly not that closely connected to hitting. Hitting (or being walked) and becoming a runner gives the player the right to progress around the bases as he chooses, subject to the risk of being put out. The safest way to do this is to rely on hits to create opportunities, but he need not wait.

What stolen bases contribute to the sport can be summarized as follows:

• By stealing bases, the batting team can improve its position without making a hit. This is the tactical function of base stealing.
• The stolen base rule gives fast players more offensive potential and thus prevents the game from being dominated by strong hitters, who may be of a body shape that makes them slower runners. At some times in the history of baseball, stronger or faster players have been more favored than than the other.
• It is generally exciting for fans to watch stolen base plays. The courage, speed, and judgment of the would-be thief are on display as are the reflexes and skills of the fielders.
• The most likely overt revision to the rules of baseball to prevent stealing bases would be to implement something like the Little League rules, which fans would complain was turning an adult sport into a kids' game. Minor tweaks to make stealing more or less likely tend to center around the balk rule and more generally stadium size and ball liveliness (which affect the likelihood of homeruns, the stolen base's true adversary).
• Like everything else in baseball, stealing has been subjected to rigorous statistical analysis. - There is a built in disincentive to steal because the benefit that the runner advances but the cost is that the runner is out. This means that a runner shouldn't attempt to steal unless the likelihood of success is on the order of 75-80 percent. But having fast players steal in favorable circumstances is part of the "small ball" strategy in which teams try to win by seizing little advantages. The bigger trend, however, is that stealing becomes less common when play focuses on the pitcher-hitter contest and at bats lead to one of the "three true outcomes" of homerun, strikeout, or walk.