TubeBuddy tells you to link your channel - so you do, then it redirects back to YouTube.com and asks you to again re-link your channel. Seems like an endless cycle.
The problem is related to the YouTube/Google+ integration. When Google combined those services it really made accounts confusing and in many cases, ended up giving people TWO (or more) accounts to select that either have the same title or same thumbnail or both.
Don't worry, there is an easy fix. You're simply selecting the wrong one account by mistake.
Click Link Channel again but this time, select a DIFFERENT account in the list than you selected last time. It will usually have the same thumbnail or same title or very similar title. If it still asks you to re-link your channel, go through the process again and select a different account this time.
If you've gone through the process and tried EVERY account of yours, but it still doesn't work, please uninstall and reinstall TubeBuddy. This usually fixes the issue.
If not, please clear your browser's cache/cookies.
If you're still unable to log into TubeBuddy, then please create a ticket here: https://tubebuddy.freshdesk.com/support/tickets/new