In particular, what is the purpose of the baking powder in this recipe:
The baking powder undergoes a chemical reaction which produces small gas bubbles in the batter. The bubbles break up the batter coating (sort of like the geometry inside a sponge) so you wind up with a mass of little holes each surrounded by a thin layer of bready material, and it all fries up airy, light and crispy; instead of one thick, dense, hard shell of cooked paste around your cauliflower bits.
Baking powder and baking soda are both leavening - they lighten the batter by adding tiny bubbles of gas.
Baking soda is sodium bicarbonate, NaHCO3. When it reacts with an acid like ethanoic acid, vinegar, it reacts to form water, a salt and CO2. The salt is fairly minute, and the CO2 provides lift.
The advantage of baking powder is that it does not need an acid - adding water to baking powder is enough to produce CO2.