A bitcoin wallet stores the private key derived from the public key/bitcoin address that it is linked to on the network. When a transaction is made from this address the wallet uses the private key to sign the transaction in order to confirm to the network that the transaction was infact made by the bitcoin address it is being sent from and then will broadcast the transaction to the nodes on the blockchain.
Additionally when a transaction is sent to the wallet, the wallet reads the blockchain and notifies the user that funds have been received
Edit: The wallet also creates/constructs the transaction before signing with the private key.