Your wallet doesn't really "send" BTC anywhere.  Roughly speaking your wallet software submits a request to update an online database (blockchain) to anyone that is willing to listen on the network.  People listening on the network (nodes) relay the update (transaction) until it is approved (added to a block by miners).  The signature from your private key verifies that the update request you submit is valid and you have the authority to modify ownership of that part of the database (blockchain).

.....I think.