Why does my dog just sit and stare at me?

Probably just because you're the center of his universe and he's hugging you with his eyes. However, there are a few other reasons. More context will help figure out which is most likely. He may want to play/walk/go somewhere with you, get attention from/cuddle with you/sleep with you, have a treat/snack/meal, trying to figure you out, or possibly not feel well.