“That's a good move on their part.”

You read a news story about Starbucks Coffee offering free Internet access at their stores. You're discussing the story with your husband. You think that this change will bring a lot of new customers to Starbucks. You say:

That's a good move on their part.

a good move

"A good move" is something you do that is smart and has a good result for you:

Buying a house would be a good move for us financially.

This phrase might be related to the game of chess. In chess, each player's turn is called a "move". If you do something that's smart and helps you win, someone might call it "a good move". So if you imagine life as being similar to a big chess game, you can understand why people use the phrase "a good move" to talk about smart decisions or actions.

a (something) on (someone's) part

When you're talking about an action, a decision, a mistake, a way of thinking, etc. you sometimes need to express who did it. You can do that using "on ___'s part":

It was a dumb mistake on our part.

This means "We made a dumb mistake."

There was no attempt on his part to misrepresent anything or to cover anything up.

This means "He didn't try to lie or hide anything."

Using "on ___'s part" sounds intellectual. You use this phrase when you're discussing or analyzing something.