Welcome to AH!
To capture a field you need to get 10 (?) paratroopers into the map room at that field. This usually means killing the field ack, as these will kill the paratroopers. You will also need to vultch, er...suppress defending aircraft, as these can also kill the troopers.
Paratroops are (currently) only available in C-47's, and can be dropped from any altitude and speed. They have a limited travel time, so need to be dropped close to the field.
The reason fuel was limited at the captured field is that some of the fuel tanks at that field had been destroyed, and had not yet regenerated.
popeye
AH Trainer