If you engage in an activity, you do it or are actively involved with it.

If something engages you or your attention or interest, it keeps you interested in it and thinking about it.

If you engage someone in conversation, you have a conversation with them.

If you engage with something or with a group of people, you get involved with that thing or group and feel that you are connected with it or have real contact with it.

If you engage someone to do a particular job, you appoint them to do it.

When a part of a machine or other mechanism engages or when you engage it, it moves into a position where it fits into something else.

When a military force engages the enemy, it attacks them and starts a battle.