What is the meaning of 能者多劳?
I would say both interpretations are acceptable.

The more works(劳动) you have to do, the busier (劳累) you would get.

A good example in real life.

My sister can't cook, so my brother in law cooks all the meals for the family.

My sister can't drive long distance, so my brother in law has to drive when they need to go to far away place.

My sister's English is poor, so my brother in law does all the talking when they have to deal with English speakers.

He has to do more works because he has more abilities; he is also busier because he has to do more works. 

IMO 多劳 means "work more" and "work more" make you "busier".

You can say 能者多劳 means "the more able man get more works and get busier".