Hi all!
As rastapete mentioned, this was a double whammy combo where a couple things weren't playing nicely for us. When 1Password's sync is set to iCloud and you're using the 1Password app from our store (not the Mac App Store) it uses a different access method for accessing iCloud and requires you login via the webpage presented in the original post. This is normal of course.
Whenever the iCloud session expires (much like any other webpage) you'll have to login again in order to keep sync functioning. To assist in this we open the login page. We aren't able to alter the login page to make things easier for you. I don't think we've gotten confirmation yet from Apple on how long it takes for the login session to expire, but when it does expire you will have to login again. What was happening was that the session was expiring every day, which made for an unpleasant experience for you all.
We're really sorry for the trouble and this should all be resolved as long as you're on the latest version of 1Password (6.3).
If anyone is still having trouble we'd love if you could contact us directly at our support forum: https://discussions.agilebits.com or via email and we'll be happy to assist however we can.
Sorry again for all the confusion and trouble you had to go through.
Kyle
AgileBits