If you dispose of something, you get rid of it. Don't want that sweatshirt with the clown's face on it? Give it away, throw it out, even sell it — these are all ways to dispose of that awful shirt.
The verb dispose comes from the Latin word disponere, meaning “put in order,” “arrange,” or “distribute" — like when you dispose volunteers to collect trash at a park. The word of usually follows dispose when it means "getting rid of something." Another meaning is "make willing or open to something," like your childhood love of reading that disposes you to becoming a life-long reader.