The LLWS 10-run rule doesn't come into play all that often. In fact, run rules in sports don't come into play very often. The teams do not engage in blowouts frequently, and most major sports leagues, and amateur leagues, don't even employ any run rules.
However, the LLWS 10-run rule is important. It's vital to ensure that the score doesn't get run up in the Little League World Series. Here's how it works.

What the LLWS 10-run rule is
If the visiting team has a lead of 15, 10, or eight runs or more respectively, the home team must bat in its half of the inning. The local league may or may not enforce this. A game determined by the 15-run rule, LLWS 10-run rule, or 8-run rule shall be considered a regulation game.

If at some point, the score exceeds a 10-run gap, the game can then be called. At that point, most games are over and the two teams don't need to waste time, risk injury and more in what is likely a decided contest.
At upper levels, teams can come back more frequently, but it's still an incredibly rare feat. In the Little League World Series, it doesn't happen enough to subject the children to what could end up being a demoralizing (even more so than a mercy rule game) defeat.
The run rule is enforced at the Little League World Series, even if it doesn't have to be in local tournaments.