This sounds like a parent feeding a baby, although I would think in AmE as well, you would be more likely to hear the child's name mentioned instead of "baby". e.g. "Come on Jessica chew it up, there's a good girl".
"There's a good girl", tells you that the child is a girl, and the phrase is being used as encouragement to get the child to do what the parent wants.
They could have used, "there's a good boy" instead to a male child.
It should be noted that this phrase is now, generally, only used when speaking to children or animals. If you do hear it used to an adult it can sound patronising, as it is often being used by one person to indicate the control they have over another. You may hear people from different regions, and mainly older people, using it to one another, where they don't think it is being patronising at all.
I have often heard it used, in the past, by a husband to a wife in the form of "Put the kettle on, there's a good girl", or a boss to a secretary "Can you type this up for me, there's a good girl". However, in BrE nowadays, this would be very much frowned upon by many. My grandparents would have used it quite happily.
I don't know if AmE sees its use, or problems of use, in the same way.