I think the idea is that cold engines run rich, so injecting air into the exhaust gives the unburnt fuel a chance to combust in the exhaust pipe before getting

I think the idea is that cold engines run rich, so injecting air into the exhaust gives the unburnt fuel a chance to combust in the exhaust pipe before getting to the catalyst. This is good for two reasons, first, it heats the catalyst up faster, which is good, and it also reduces the amount of unburned hydrocarbons.