Yeah, I'm really enjoying how little AirDroid seems to give a damn about fixing anything, because apparently this bug has been going on for a month or more.
What's happening is the window is trying to close the login dialogue but it's coded badly so it just closes the whole window. The only way I've found to get around it is to do it in Internet Explorer so when it says "this page is trying to close the window, allow?" you click "no" and then you can access the app.
Also, the Airdroid MAIN page is so full of errors it took me four attempts at registering before it worked, because it kept shooting authentication errors at me every time I tried to connect with Google or Facebook, so now I can't use my actual email address because it halfway-registered with NO PASSWORD. Ugh.
Airdroid, y'all made a good app, but it's like you just stopped caring.