He's talking about machine gun ammunition capacity. It was really only a few minutes worth if you held the firing buttton down. This is one reason pilots were trained to fire in two to three second bursts (the other reason being that bursts were more accurate)
Using the Browning .50 Caliber Machine Gun as an example (since it was one of the primary weapons used on USAAF aircraft during the war) the rate of fire was 450-600 rounds per minute.
The P-51D Mustang had six fifties on board and carried 400 rounds for each of the two inboard guns, 270 for each outboard gun. Also consider the fact that, on average, every fifth round was a tracer (approx. 450 total). As you can see, that doesn't leave much actual trigger time before all of your ammunition was expended.
One reason for the minimal amount of ammunition onboard was the obvious factor of the weight; heavy planes go slower and thus are easier targets to hit. As well more weight equals more fuel consumption and the longer a pilot can loiter in an area the longer that area remains under his control.