I understand why the air is pumped into the cat, but why does it go into the headers, too?

 

Same reason, to aid in oxidizing unburnt fuel. Up at the exhaust ports, the exhaust is as hot as it's going to be, so injecting the air there will have the maximum possible effect.

Until feedback computer controls came into being, the A.I.R. was injected at the manifold only.

A.I.R. existed before catalytic converters.