same issue here (MB Pro Retina,OS X 10.9.1 (13B42).
With the first time toolbar launch I got sign-in window. After typing my credentials and click "Sign-in" button nothing happens. When I quit the application and launch it again I get only "Connecting" message and after 10 sec. "Could not load data from Tootledo website!"
I have Little Snitch Network Monitor installed on my Mac. I can see a traffic to m.toodledo.com during "Connecting" stage. Captured that traffic to pcap file and opened it in Wireshark. I can see HTTP connection, TLS handshake and application data over HTTPS.
It seems that there is no issue with the connection.
Tried to remove the app and install it again. Still the same problem.