Engaged means fully occupied or having your full attention. An engaged reader really focuses on the words and maybe even jots down questions or comments in the pages' margins.
If you're busy or involved with something, you're engaged in it. A formal reply to an invitation to lunch might be, "I'm sorry, I'm otherwise engaged" — meaning you already have plans. Engaged often describes people who have committed to getting married. You can see this meaning in the word's origin, the French word engagier, meaning "to pledge."