I know that in a transaction, the difference between the amount and the sum of the outputs is the implicit transaction fee, but how do the miners that verify the block containing that transaction actually get access to that fee as UTXO?
One of the network rules is that a coinbase's outputs cannot be larger than the current block reward plus the transaction fees of the transactions in the block.
So, a miner claims the transaction fees by increasing the amount of the block reward. This is true even after 2140, when there will be no block reward. (Network rules require that the first transaction be a coinbase, even if there is nothing to claim.)