Engage means to bind, catch, or involve. If your sink is stopped up, engage, or hire, a plumber to fix it. Otherwise the smell of rotten food in the garbage disposal will engage your attention (in a bad way).
Engage comes from a French word for pledge. If you agree to marry your sweetheart, you are engaged to marry. If you engage the kid next door to water your plants, she will expect to be paid for it. Engagement is also used to talk about war or conflict. An army engages its enemy in battle, just as you might engage in debate with a traffic cop over whether the light was yellow or red.