The offer button on auctions is a way for a buyer to get the item for a price they're willing to pay without either the buyer or the seller having to wait out the 7 or 10 day term of the auction.
When the seller has an auction listing, there are several options from which they can choose. (Seller can't have a BIN and an offer. Seller must choose one or the other.)
1. Auction only (no BIN)
2. Auction with BIN
3. Auction with make an offer
1. With auction only, bidding starts at opening price and wherever it ends, that's the selling price.
2. With auction with BIN, bidder can either bid or BIN. The benefit to the buyer in bidding is that they might get the item for less than the BIN price. But the bidder/buyer also runs the risk of the bidding going over the original BIN price
3. With an auction with an offer option, the seller starts the bidding at the lowest they're willing to accept. But they give a buyer the ability to end the listing early if the buyer makes an offer over the starting price but at a fair deal for both the buyer and the seller.
With this option, it's the buyer who is setting the BIN price rather than the seller.