Of course they can change hands outside the ledger. We could create paper wallets and exchange them on the street and the blockchain isn't going to know that ha

Of course they can change hands outside the ledger. We could create paper wallets and exchange them on the street and the blockchain isn't going to know that happened. A tumbler is just a more complex version of that. Imagine this scenario: You have 10 BTC of illegal funds you'd like to launder. I, and 3 of my friends each have 5 BTC and they're all wanting to do the same. I transfer some money into a new wallet from 2 of my friends and give it to you. Your money goes to me and 1 of my other friends. This way the blockchain has no link between your source funds and the money that you received. This is a simple example. If you split the money into random amounts and do this a few times with new wallets each time it can easily become untraceable.