This is one of those 'funny' sentences that are technically correct but no native speaker would ever use, much less understand if spoken quickly.

To understand it, break it up starting from the outside.

The rat ate the malt. 

The rat (that was killed by the cat) ate the malt.

The rat (that was killed by the cat (that was chased by the dog)) ate the malt.

Usually no one would use this. People would have no problem saying 'The rat the cat killed ate the malt', but once you add a third embedded sentence it becomes very confusing. 

Thank You Jack

l Understood that, good explanation.