If there is a asteroid on a collision course with Earth, and it's big enough to cause some catastrophic damage, everyone on the planet will want some plan in action to make it go away. The simplest way to do that is to be able to do that is to alter the path of the asteroid ever so slightly that its course misses the Earth. Even though a satellite has very little mass, if you make it travel fast enough, you can impact enough velocity to give a big asteroid a nudge. Even if it's a tiny bit, that redirection propagating over millions of km will save the planet.

DART's mission target was actually a smaller space rock (roughly the size of the pyramid of Giza) that was orbiting a larger asteroid. Since they could easily detect the smaller rock's orbital period, they could calculate how much the impact would change its orbit. This would better inform scientists about what to expect from future redirect missions so that we could make a robust planetary defense system in case The Big One ever shows up.