Quote from: the FAQ

How does Bitcoin work?

Bitcoin utilizes public/private key digital signatures (ECDSA). A coin has its owner's public key on it. When a coin is transferred from user A to user B, A adds Bs public key to the coin and signs it with his own private key. Now B owns the coin and can transfer it further. To prevent A from transferring the already used coin to another user C, a public list of all the previous transactions is collectively maintained by the network of Bitcoin nodes, and before each transaction the coins unusedness will be checked.

In addition to hashing out new blocks, all of the transfers on the network need to be verified - so that's what your client is doing.

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