A transaction is only an envelope for the signed bundle that details the exact transaction of funds between addresses. The transaction gets attached to the Tangle by doing the Proof of Work and validating two tips. Once it is linked to the Tangle that way, it cannot move any more. It therefore has become part of the permanent history of the Tangle. It can only get cleaned up by a snapshot but will still exist on perma-nodes.
When you reattach a transaction, what happens is that you take the signed bundle out of the original transaction envelope, and create a new transaction envelope with that same bundle in it. You then will have to do Proof of Work again, and validate 2 tips, to attach the new transaction envelope to the Tangle.
Note that you now have two copies of the exact same bundle in the Tangle, but with different transaction envelopes, each with a different transaction hash. That is why a reattached transaction shows up as a new one in your transaction history.
Also note that it does not matter who does the reattaching. The sender can do it just as well as the receiver of the transaction. In fact, anyone may do it.
This mechanism separates the Proof of Work and validation of tips from the signing of the bundle. This is how the mobile wallets are able to function. They pass the attaching of the bundle to the tangle (and thus the PoW) off to a node that will do it for them.