Certainly, signing a transaction offline without broadcasting it is like doing nothing.
However, signing a transaction offline is a great way to keep your private keys safe. This is why Armory is considered so secure. You can broadcast a transaction online without signing it online. You mentioned that broadcasting a transaction requires you to be online, and this I agree with. The part you failed to mention (and that I added) was that you don't have to sign the transaction on an online computer.
No need to Google. I'm pretty sure I understand the basics of bitcoin transactions at this point.