In order to do this you must already have more than one wallet for any one asset. Once there are multiple wallets created, the 'Transfer Within Wallet' button will no longer be grayed out.
The 'Transfer Within Wallet' icon will be grayed out if there is only one wallet that exists on your account.
Once you click that button you will see the transfer panel. Select the origin wallet, the destination wallet, and the amount you wish to transfer. Click 'Confirm' to instantly process the transfer (the page may require a refresh after you have confirmed it).
Sending 300 BTC from Main BTC Wallet to BTC-Wallet2