Bitcoin didn't split in half, it's following did. Even that is technically wrong since it seems both are getting good usage, I expect one to die eventually though. Either way, if you had BTC before the fork (the name for splitting) you have both BTC and BCC.

All transactions on the blockchain refer to an address and can only occur if the transaction is signed by that address's specific private key. As long as you have the key you have access to the funds, including transferring it. Your balance of BTC comes from adding up all transactions involving you and seeing where you end up.

In a fork, the list of transactions (ledger) is duplicated creating two coins. Every address now has both BTC and BCC. In order to access the BCC you need the private key for the account that owned BTC before the fork. If you held your BTC on an exchange it's up to them whether you get your BCC which is why I always suggest having the private key for anywhere you store your crypto. Etherdelta is a great exchange for that, as is any decentralized exchange.

What's nice here is that your btc and bcc wallet have the same Pkey so you only have one to protect. Although if you lose the one PKey you lose both accounts...