Here are a few, off the top of my head:
Often there are factors that you may not think are relevant when asking your question, but could certainly be. You may have looked up some kosher-kitchen question about vegetables, not knowing that onions have very different laws than potatoes.
There are plenty of gray areas in halacha where the conclusion may be something like "it's best not to do such-and-such, but it's allowable in cases of great need"; if every Tom, Dick, and Harry would go around deciding for themselves what's great need, X% of the population will (wrongly) do it all the time, and Y% will (wrongly) refrain even when it's life-and-death.
Someone other than yourself may be the best judge of the situation. You may say "it's not a great need", but a good rabbi may hear the anxiety in your voice that implies otherwise. Or the question may involve other human factors, where knowledge of the people involved is important.
It provides a good "checkup opportunity" for all sorts of related issues. If a married couple brings a personal question to their rabbi, he can see if the interactions between them look healthy. If a man needs to shed his yarmulka or eat fruit at a non-kosher restaurant because of his job, it's important to make sure he isn't feeling alienated from Judaism.