How does activity measure flights climbed?

It used the combined data from the altimeter and the accelerometer. If it used the altimeter alone, it would record that you had climbed stairs when you rode in an elevator.

As noted, it's pretty reliable. However, if you're hiking and doing lots of short elevation gains or loses, it can sometimes count those as flights of stairs. As they are, from a fitness standpoint, pretty much the same, it's not something I worry about much.

In the Health app it defines a flight of stairs as:

"A flight of stairs is counted as approximately 10 feet (3 Meters) of elevation gain (approximately 16 steps).”

The measurement uses motion sensor data and barometric pressure data to determine the distance walked and to differentiate it from flights up in an elevator for example.

Apple Watch use the built-in barometric altimeter to count flights climbed based on atmospheric pressure changes. Its not 100% perfect but usually not too bad either.