If you have your old chain, you can measure the rivet spacing on 12 links. Ideally, it should measure 12" (brand new). As wear occurs, this spacing will increase. If the same length measures 12-1/16" the chain probably did not damage the cassette. If the spacing measures 12-1/8", then chances are the cassette was damaged by chain. I tend to measure my chain as a habit, just to avoid damaging the cassette.
Basically, a worn chain physically damages the cassette by forcing the cassette to match the chain dimensions. Then when you install the new chain, it doesn't match the worn cassette and hence chain skip.