Both units come from the inch
A cubic foot is 12 × 12 × 12 = 1,728 cubic inches, and a US gallon was defined in 1707 as exactly 231 cubic inches. Divide one by the other and a cubic foot is 7.480519… US gallons — an exact ratio, not a measurement. In litres a cubic foot is 28.3168, and an imperial gallon (4.54609 L) fits 6.2288 times, which is why UK and US tank capacities never quite agree.
gallons (US) = cubic feet × 7.4805194805 | cubic feet = gallons (US) ÷ 7.4805194805
Where this comes up
Refrigerators and freezers are rated in cubic feet (a 20 ft³ fridge holds 150 US gallons of air). Water meters in many US cities bill in hundreds of cubic feet: one CCF is 748 gallons. An aquarium 4 ft × 1 ft × 1.5 ft is 6 ft³, about 45 gallons. For a pool, multiply length × width × average depth in feet, then convert here.