Why does cake batter rise again in the oven, long after mixing?

Because most baking powder sold today is built to act twice, on purpose. Baking powder is a dry chemical leavening agent, a mixture of a carbonate or bicarbonate and a weak acid, kept apart by a drying agent like cornstarch until water arrives. It works by releasing carbon dioxide gas through an acid-base reaction, and those bubbles expanding is what leavens the batter.

The trick is that the acid in modern baking powder can be either fast-acting or slow-acting. A fast-acting acid reacts with baking soda as soon as the mixture gets wet, at room temperature, while a slow-acting acid waits until it's actually heated. When a baking powder uses both kinds of acid together, it's called "double-acting", and this is the type most people buy today. That's why batter puffs up a little in the bowl, then rises again once it hits the oven's heat: by providing that second rise in the oven, double-acting baking powder makes the timing between mixing and baking much less critical, so a few extra minutes on the counter won't ruin the cake.

This two-step trick took a while to arrive. A rudimentary chemical leavening was used in the United States around 1790, but it left a bitter aftertaste and reacted so fast that the dough had to be baked right away. The first commercially sold baking powder, Borwick's, launched in England in 1842, after George Borwick spent eleven years perfecting a formula from his father-in-law, a chemist, as a yeast substitute. A year later, pharmacist Alfred Bird developed a single-acting version that released all its gas the moment it got wet. It was Eben Norton Horsford, a Harvard professor, who in the 1860s developed the first double-acting powder, the kind most widely sold today.