Why does transaction activity show on the blockchain for my Bitcoin address at my hosted (shared) E-Wallet service? I have an online wallet at localbitcoins.com (address 12pHFR5yv2KGqck5N8teb5MgoRtzAgLRn2).

I made three transfers to it, for a total of 0.5 BTC + 0.5 BTC + 0.3031 BTC = 1.3031 BTC.

Blockchain.info did show my address receiving coins. However, it is showing my address sending coins to several addresses after receiving them https://blockchain.info/address/12pHFR5yv2KGqck5N8teb5MgoRtzAgLRn2:



Is someone using my online wallet at localbitcoins.com to withdraw money?
You do not have a Bitcoin wallet at LocalBitcoins.  Instead you have only a ledger account with their hosted (shared) Bitcoin E-Wallet.  

What this means is that you cannot use the blockchain to track your balance.  You can only use it to determine how many confirmations your transfer (deposit) has received.

The E-Wallet provider simply credits your account when they see a confirmed deposit transaction (e.g., 6 block confirmations) to a Bitcoin address associated with your E-Wallet account.

To know your balance, you will need to login to your E-Wallet account on LocalBitcoins.  That will show you your balance.  The account details will show deposits, withdrawals, purchases and sales.