“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 say this because you think that this change will bring a lot of new customers to Starbucks.
That's a good move on their part.
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"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.
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.