Just tried on another machine – same results.
I click the connect button, a small window appears for the username and password, enter my info and press Log In, button says “Please Wait” for a few seconds, small window closes, main page appears to be reloading, and ends with the same connect button.
The problem seems to be something with WordPress 3.8. I just tried the plugin on a WordPress 3.7.1 install and while the login process appeared to get hung-up, I was able to go back into the mailchimp page through the dashboard and I was now logged in.