Why must some aircraft get permission to start their engines?
Airliners usually push back from the gate and then start their engines when well clear. 

If the aircraft has an unserviceable APU, a ground based power supply must be used at the gate for the start. Permission(coordination) is required so that their are no hazards for ground crew and vehicles in the gate area.

After one engine is started at the gate, the aircraft is pushed back. The next start, still needs permission(coordination) because the operating engine usually has to increase it's thrust above idle in order to have enough bleed air pressure for the "Cross Bleed Start".

EDIT:  "Permission" may be not be mandatory at many airports, so "Coordination" with Apron Control, Ramp Control, or Ground Control may be a better description.