How are transaction fees distributed?
Each ~10 minutes, a "block" containing transaction gets crafted and broadcast. Miners choose which transactions to include in their block -highest fee/Kb transactions, to maximize their earnings- and they collect the transaction fees of the transactions they've chosen when they find and publish a block.

Consider this transaction:



You can see the:

Included in Blocks: 499089


If you browse to that block in the block explorer...



So, the transaction fees of that transaction are collected by the mining pool ViaBTC.