What's the usage of 고 있다 in this case? The normal usage is to state that the act mentioned in the preceding statement is continued but it doesn't seem fit this example.
Here, 놀고 있어 is not a declaration, but a command: it's the same as 놀고 있어라.
So the meaning is something like "Go keep playing over there."
There is not much practical difference between 놀아(라) and 놀고 있어(라) here - after all, mom is telling the kids to keep playing - but 놀고 있어 sounds softer to me, which fits the situation better. Essentially, mom is saying "Thanks, but it's OK if you keep playing."
In other context, there can be a difference in meaning: e.g., if I say 공을 던져, I'm asking to throw the ball, presumably once. If I say 공을 던지고 있어, then I'm clearly asking to keep throwing the ball (or balls) multiple times.
있다 can be used as an auxiliary verb behind another verb, like -고 있다. It means that the action that the front verb indicate is continuing or the result of the action keeps going.
So when the mom said "놀고 있어", she told her child to continue playing.
먼저 먹고 있어. 금방 다녀올게. Please keep eating this first. I will be right back.
TV 보고 있어. 거의 다왔어. Please keep watching TV. I'm almost there.