This is the magic of the blockchain.
The reason only one copy ends up on the chain is that when you sign a transaction and broadcast it over the network, it is picked up by miners who verify the signature and nonce.
A nonce is a number that essentially represents which number transaction of an account this is (first, second, third, etc.). In order for a transaction to be valid, it must have a nonce that is exactly 1 more than the last transaction sent by that account in the blockchain.
The miners each arrange all of the valid transactions they receive, making sure that nonces are strictly increasing. This is why the transaction is not included over and over, because two transactions from the same account cannot share the same nonce.
The miners then compete to solve a difficult problem, proving that they contributed a certain amount of work, and the one who finishes first gets to publish the block and claim a reward.
Once the block is published, each node verifies it themselves, to make sure that the miner was honest.