The Quick Connect Option (Tools > SSH > Quick Connect) lets you access remote servers on an ad hoc basis. You just enter the remote server credentials : user name, hostname and optionally port.
Specify a friendly name if you want to save the connection details.
If you do not want to install a public key remotely, then Codio will connect to the remote server, which will prompt you for a password.
When you press Connect, a tab will open up in the current panel and off you go. The connection details will not be saved.
Connect & Save
This connects but also saves the connection details to the SSH Connection Manager.
If you want to add your Codio public key to the remote server, then you should Connect & Save. Then go to the SSH Connection Manager and press the public key button as described here.