How to control 120V power outlet with low voltage
A relay, just like the other stuff your thermostat controls uses for this purpose.
Not surprisingly, you'll find that 24VAC coil 120/240VAC contact relays are very, very common, and available in a wide range of ratings. You'll need one rated to operate the horsepower of your fan motor (or more, but not less) at 120V. It may also be rated for more horsepower at 240V, but you need HP at 120V rating if the fan motor is wired for 120V.