1. To use the airstrike, you have to be in first person - not on the map!
2. Scroll down to Airstrike (or Apache or Artillery, depending on which mission it is) and point your crosshairs at the target. (For me it's been enough to point at the top half of the radar dish on top of the AA-battery to call it in).
3. While Airstrike is selected, call up your Crosscom menu - wherever you have that mapped - and choose the Attack option, still pointing at the target.
4. Take cover and enjoy the fireworks!
HINT: Try using the airstrike as much as possible on any other targets before you reign fire down on the objective's targets. The airstrike/apache/artillery will dissappear as an option the moment you've destroyed all the targets that are part of the objective.