Let's say you have 0.5 BTC in your Bittrex wallet. On the day of the fork you will find out your BTC wallet still has 0.5 BTC, but you will also find that you've got another wallet, a BCC wallet with exactly the same amount of 0.5.

The best way to think about the fork(for me) is equating the blockchain to a train track on which there is a train filled with passengers each with their own seating arrangements(wallets). The train reaches a fork in the train track. At that moment the train duplicates itself and 2 identical trains carry on on both tracks after the fork. Until that moment all passengers still had their own seats and the train driver knew where each would move if they moved. After the fork each train driver knows about their own passengers, but not about the ones in the other train and each train will develop independently from the other. Bittrex could be imagined as a train station operator. It already had a station for loading/unloading the passengers from the original train(BTC) but it is also opening a station(wallet) for the other train (BCC).