How does the bitcoin transaction fee get paid to miner(s)? According to https://bitcoinfees.info/

The current "cheap" rate for a bitcoin transaction is $22. 

How does this get paid to the miners? Is it just one miner that verifies the transaction, puts it into the next block, and collects the $22? 

I understand proof of work. But I don't quite understand how a transaction fee gets paid out to a miner. Thanks.
The very first transaction in every block is called a coinbase transaction. This coinbase transaction is created by the miner themselves, to pay them the block reward and the fees from the transactions in that block. At the time of writing, the block reward is 12.5 BTC, and then the total of all the fees is added to that, and the coinbase transaction is created to pay themselves that total.