The publisher has been through its rounds of proofs, and has given the printer the go-ahead to print the entire run of books.
Now we take a symbolic pause, to represent the three months or so the printer will take for this.
(No, printing even a gigantic run of books is the work of days, not months. But it goes in a queue behind hundreds of other books.)
Finally! The books are coming off the presses, all bound and whole and new! Real books!
Most of them are loaded onto giant pallets and sent down to the sea, to be put on a ship across the Pacific. This, the main shipment, will reach the publisher's warehouse in about a month, and then the warehouse will start filling orders to bookstores.
But a couple hundred copies are sent air-freight to the publisher, and these are advances, because they come in advance of the main shipment. They are for marketing. As with ARCs, they represent an extra cost to the publisher (because books are heavy, air-freighting them is not cheap). So, again, please do not make any assumptions about how marketing's advance copies should be shared with the outside world. They have many clamoring sales people who need a copy, and of course reviewers and magazines and places like that to send the advances.
Your editor may get a couple advances to send to the author and illustrator. But the gratis copies mandated in your contract will not come out of the advance shipment. You will have to wait till the main shipment is in the warehouse, and then you may need to remind the publisher to send them (if you want it to happen promptly).
Now, advance copies are not to be confused with copies advanced, which, I recognize, is asking a lot of the uninitiated.
Copies advanced refers to the sales of the book in its first three months. This is a bit of a thermometer for how the book is being received in the marketplace, but how that thermometer is read varies a great deal from house to house, so I won't get into any speculation here. If you have a conversation with your editor and this term comes up, ask her what practical significance those sales numbers have at her house.