The home run in this context means that he is coming to the end. Like he has gone out for a long run and now he is on the way back and getting closer to home. So it means he getting close to finishing the house. In athletics, the the last part of a track race, after the final bend, is called the home straight, which can also be a synonym for the home run.

A home run comes from baseball. This is where the batter hits the ball hard and attempts to run fast around all of the bases. The meaning in this context is that Dan is running fast to complete a task; the same way a batter wants to run around all of the bases in a baseball game.