I suspect "catting away" is a typo, and the original was "chatting away". That would be a very common, idiomatic way of saying that he was talking a lot, and I've never once heard "catting away".
Also, although "the drive" certainly could refer to the driveway, that doesn't ring true to me. I reckon it means the time they were in the car together, driving from one place to another. It doesn't necessarily imply he was driving at all.
Also, I respectfully disagree with Francisco - "almost continuously" would be quite a common way of saying that he was talking a lot.