How to know when to change a chain? A bike shop employee used a tool he said would check the wear on the chain, and said that it should be changed. 

He said it took a measurement that meant the chain had expanded, but it is still hard for me understand why I should change the chain. 

He said that the reason was that the expanded chain links will cause the chain rings to wear down. 

That is one argument and I have no experience with it, to know if it is true. 

But are there more reasons? How do I know when the chain should be changed? 

My first thought is to use it until it breaks, but it probably is not the wisest option.
You can measure the stretch of a chain with a chain gauge (or just a ruler) 


Alternately - the chain links are 1 inch long, so measure 12 of them with a ruler, if they are more than 1/8inch longer than this then they are worn.

You need to do this before it wears the rear cogs - it will cause the teeth to wear into sharp pointed spikes = need a new cassette.