There is a capacitive (electret) mic and air blows past it, and some models have a ring that adjusts the airflow non-electronically. A chip on the mic’s tiny PCB crudely measures sound level (which is way higher when breathing as opposed to any other sound) and drives its power transistor to switch the heater. It probably also refuses to work below 2.5 V to prevent the battery from discharging too deep. Some models also drive an LED with their heater output.

Musicians use breath controllers to play digital wind instruments but those are obviously more sensitive and complicated (likely using a piezoelectric or MEMS pressure sensor on a membrane of a chamber the user breathes into).

Here's a video of it in action (turns out it flashes for a while after you stop breathing) and the best photo I could take with my phone and two magnifying glasses (sadly, the trick of a water droplet over the camera didn't work because of its hydrophobic coating).