How are transactions verified to ensure only the eligible person can spend a Bitcoin balance? I know each user has a public key and private key. The users signs the transaction with his private key and sends it to a Bitcoin address. What prevents me from claiming his coins? I read something about verification with the private key but I didn't really understand. How does Bitcoin ensure that only the eligible person can spend the bitcoin? More specifically I didn't understand the verify and sign part in Satoshi's paper in the diagram on page 2.

Also from this diagram, how is it that a secret key can act as a key to lock a lock, but the public key can open the lock only but can't lock it back? How is this implemented technically?
Read about Public-key cryptography and Digital signature.

For bitcoins you need the private key to spend (by signing) the coins that were "sent" to the address associated with a public key.