How can the blockchain tell that a transaction was really sent by you?
A signature is bound to the specific content it signs. If even a single byte of that content changes, the original signature will no longer verify. Ethereum transactions include a chain ID (EIP-155), so a signed transaction cannot be replayed on another chain.
A digital signature proves two things at once: the transaction really came from the owner of the private key, and its contents have not been tampered with after signing.
When transferring $BTC , the wallet signs the transaction with the private key, and nodes across the network verify the signature using the public key. Current: 81888.01.