Someone who's preachy will give you self-righteous advice whether you ask for it or not. The worst movies are the ones that have a preachy moral at the end.
When you're preachy, you feel morally superior to other people and you're not afraid to talk about what you think they should do. A preachy neighbor might complain about how rude the kids on his block are and suggest that their parents should teach them some manners. A preachy children's book might have a patronizing message about always saying "please" and "thank you." Preachy is an informal adjective that dates from the nineteenth century.