eosjs provides an example implementation of the SignatureProvider interface called JsSignatureProvider.
Although JsSignatureProvider is insecure and should not be used in production, it provides a basic example of what a SignatureProvider is and does. JsSignatureProvider simply takes a list of private keys as strings in its constructor and maps these private keys to their respective public keys. When JsSignatureProvider's sign is called, a buffer of the chainId and serializedTransaction is created and eosjs-ecc's ecc object is used to sign the buffer with the private key corresponding to the required public key.