I'm trying to wrap my mind about how transactions are secured. I've read through the wiki and I grasp how inputs/outputs work, I'm still confused about how signing does.

This is a representation of a bitcoin transaction that I've seen all over the place:
http://www.techinasia.com/techinasia/wp-content/uploads/2011/07/bitcoin-transaction.jpg


Since, I believe, most transactions are sent to a bitcoin address so the recipient's public key is not known in advance, how does the recipient claim the coins?

To use a reddit-ism, please explain it like I'm five