Lots of good answers so far. From my experience, most people know what the fuck to do or at least they know they have a few options, so they don't need me to tell them how to solve their problem.

To specifically answer your question... sitting there listening allows you to figure out if the person needs to vent, talk out their options, validation about their reaction, validation for their own solution, social connection, to know they have a support system in you, encouragement, and sometimes, yes, advice or a solution. So to assume it's only a solution they want leaves out a whole host of other possible reasons they might come to you. "I know I just need to send an email, I just wanted to removed about my boss being a bag of dicks".