The instructions are actually pretty simple. Just go to the Developer Area in LinkedIn, below:
www.linkedin.com/secure/developer
Click into your App and somewhere, there should be an "Upgrade App" or "Migrate App". I don't recall the specific setting offhand, but it should be pretty recognizable. If you can't find it, let us know.
Once done, when user's authenticate on your site, they'll see a different popup which will specifically note that your app will be getting their email and (possibly) some other profile information depending on what you're importing. If you previously had some LinkedIn authentication on your site, your users will likely need to re-authenticate with the new popup. All other features should remain the same.
Obviously, if you were using your app for 14 months, please test everything to make sure the upgrade doesn't affect anything badly, but it should be very straightforward.
If you run into issues or are confused about anything, just let us know.
Thanks,
Alex