The immelmann is essentially a maneuver for repositioning.
Not to be confused with it's WWI namesake, its main value lies in enabling the fighter to reposition at any angle with almost no lateral displacement.

Back in 1916 the original immelmann turn was more akin to the vertical reverse than its present-day counterpart.
The modern version of the immelmann is a vertical climb or half loop, possibly aileron-turning during the climb, then rolling out into level flight at the top.
Its main value lies in using the vertical plane to change the direction of flight in the smallest possible horizontal space.
Horizontal turns at normal fighting speeds take up a lot of room laterally.
Using the vertical plane enables the fighter to turn square corners in relation to its position above the ground.
This maneuver makes repositioning for a further attack, or to meet a threat, much easier than would be the case using horizontal maneuver only.

The immelmann is basically a half loop with a half roll on top.
After you've passed your enemy, pull back gently on the stick, apply full throttle, and start climbing directyl up.
Make sure you have plenty of speed. Meanwhile, look behind you to see what the bandit is doing, and start planning your turn toward him.
Before you enter a stall, do a half-aileron roll toward the enemy and look for the chance to drop in toward him.
If you time it right, you'll get on his six o'clock position and should be able to get a decent deflection shot at him.

Drawbacks:
Because of the loss of speed toward the top of the maneuver, you should avoid using the immelman when enemies are within gun rane; you'll be a stalling duck if you're not careful.