So I'm guessing that being an offline wallet, it can't send transactions by itself, only generate them. So how does the "pairing" with the other wallet software work for broadcasting such a transaction?
The AirGap Vault signs a transaction and provides a QR code for the signed transaction. The wallet to which it is paired (e.g AirGap Wallet, Metamask, Sparrow, Spectre, Rabby) scans the QR code and broadcasts the transaction.