PROBLEM: New user buys BitCoins and then starts spending them, but at one point does not copy and paste the full BitCoin address into the client, or copies the wrong BitCoin address into the client. The result is that BitCoins are sent into the gutter, never to be retrieved.
SOLUTION: Wrap the send packet with the receiving BitCoin address, so only the receiving BitCoin client can un-encrypt the packet and transmit the packet back out to the network. Thus, the sender can be assured he is sending it to a real client.
If you don't paste the full address, the checksum will stop it. If you do paste a full address, but it isn't the address you intended to send to, won't the (wrong) receiver just decrypt the packet?