My question is, where/how is that public address associated with my private key besides on the piece of paper I printed?
This is really the heart of cryptography - As demonstrated by rumbilita, the random input (step 1) is used as a basis for your private key (step 2) and this is used as a basis for the public key (step 3). There is a consistent method to produce the private key, which will consistently match to the public key. It is not possible for anyone else to reverse the method to produce the private key, so your private key alone is sufficient to grant access to the bitcoin wallet.
The only trap (as mentioned in the article) is if you don't use a unique input to produce the private key. Aside from that, there are very slim chances that another person could produce the same private/public key pair.
Suppose I am offline, with a computer that loaded bitaddress.org but have no access to the internet at all. I generate/print a public/private key pair. Then, on another computer, I send 1 BTC to the public address. At this point I have "loaded" my paper wallet correct? That 1 BTC is now in that public address.
Correct, at that point the bitcoin blockchain has recorded a transaction of 1 BTC to the public address. At any time in the future, you can submit another transaction to send that 1 BTC to another public address, and all that is required to authorize the transaction is your private key. That is why protecting your private key is imperative, if someone gets your private key the first thing they will do is send all the bitcoins out of your wallet to a public address they control.