What exactly is the transaction fee? The fee of finding a block? The fee of transferring the btc reward to another account?
Transaction fee is a method by which anyone who creates a transaction incentivizes the miners to include that transaction in the block that the miner will mine. Bitcoin blocks can have a maximum 4MB of block weight, which means there are only certain transactions that can be included in the block that is being mined. So you include a fee so that miner picks your transaction over others to include in the block. Transaction fees also serve another purpose of preventing DoS attacks. If there were no fees, people in order to spam the network would send between between addresses they control so as to deny the service to legitimate people. Transaction fees ensure that there is a cost associated for launching such attacks.
To be more thorough, you do not send a transaction fee to a particular address, but it is the difference between the input and output amounts. So consider a transaction that consumes 1BTC and sends 0.8BTC to one address and 0.15 BTC to another address, then the transaction fee is 0.05BTC. When the miner creates the coinbase transaction for the block he is mining, he will pay out the block reward (12.5 BTC currently) and sum of all the fees of the transactions that are included in the block to the pool participants.
As to your other question about how they pay transaction fees vs block reward, that is more of a service related question and is more suited to the customer relationship department of each pool.
