Yeah, they used to use film reels (not video). The way he described it, it was similar in principle to Nintendo's Sky Hawk from 1976:

https://www.youtube.com/watch?v=XLcT5eaUZw0

This is the actual film footage used:

https://www.youtube.com/watch?v=1-lUgkT2sz4

I don't know if the game my dad remembers worked the same way, but as you can see the footage is divided into two halves: the top half is the footage of the plane flying, and the bottom half has both a spot of light and footage of the plane exploding. A mirror inside the machine normally is positioned such that it reflects only the top half of the picture to the screen. The spot of light on the bottom half corresponds directly to the plane's location in the top half; a light sensor corresponds to the gun's motion such that when the gun is pointed at the plane, the sensor will pick up the spot of light (similar to the NES Zapper) and register a hit. Then at the right moment, if a hit was registered, the mirror will rotate, shifting the image from the top half of the film footage to the bottom half and showing the explosion sequence, before resetting for the next plane.