Why does baking soda make things rise?

Answer: It releases carbon dioxide gas when it reacts with acid or heat

It absorbs moisture and swells up — Baking soda dissolves in moisture, but dissolving is not swelling. It makes things rise by producing new gas, carbon dioxide, through a chemical reaction; without an acid or heat to trigger that reaction there is no lift.

It releases carbon dioxide gas when it reacts with acid or heat ✓ — Correct! Baking soda is sodium bicarbonate, a base. Mixed with an acid in the batter, such as buttermilk, yogurt, vinegar or lemon juice, it reacts at once and gives off carbon dioxide; strong heat also breaks it down on its own (2 NaHCO₃ → Na₂CO₃ + H₂O + CO₂). The gas collects in bubbles trapped by the batter, and as they grow the mixture rises and sets with a light, open texture.

Heat expands the air already trapped in the batter — Heat does expand the air and steam already in a batter, and for some things, such as popovers and puff pastry, that is the whole story. But the small amount of air mixed into a cake batter cannot swell it to double its size; baking soda matters because it creates far more gas than was there to begin with.