This 90 second video explains how Bitcoin transaction fees work: How Bitcoin fees work.
What the video covers
When you send a Bitcoin transaction, a fee is paid to Bitcoin miners, which gives them the incentive to include your transaction in their block. The fee is paid in bitcoin, and it is up to you to decide how much to offer.
Getting a transaction into the next block is competitive, so the more you offer, the better the chance it is picked next. Before choosing a fee, check a block explorer such as mempool.bitaroo.net to see how busy the waiting list (the mempool) is.
Historically, very high fees have lasted only for brief periods, so waiting for fees to come down has been an effective strategy when a transaction is not urgent.
Fees are the only proven way to prevent spam on the network without a central authority, which is why Bitcoin has them.
A withdrawal fee is a different thing. The fee you pay an exchange to withdraw your bitcoin goes to the exchange, which then creates the transaction and pays the miner from its own funds. For the options on Bitaroo, see BTC Withdrawal Fee.