There is no "activation code" on the picasa site -- I have no idea what these instructions refer to. Here's what finally worked for me (taken from a Google forum post):
1) Internet@TV from the main menu
3) System Setup...here is where we think we set up our username and PIN
4) After setting up service site ID, go to Manager (under Internet@TV) and choose "service site" - it should ask for your PIN
5) Select Picasa and register your existing Picasa account login info (you must have a Picasa account set up first)
6) Return to the main Internet@TV menu and choose Samsung Apps
7) Select "Login" using the red "A" button on remote, and enter your PIN again
8) Choose Picasa (under "What's New" or "Lifestyle") and select "Run Now"
9) This should connect you to the main Picasa app, now connected to your account
Hope it works for you, too!