Note that Splashtop Business client apps must be installed for this to work (version 2.5.5.10 and above for Windows and version 2.5.5.5 and above for Mac).
Download the latest Splashtop Business client app for your platform from www.splashtop.com/downloads#b.
If the client app is not installed or if it's an older version, connecting from the web console will result in a browser error page or have no response.
How does it work?
On the web console, there is a "Connect" button next to each computer. Click the button to connect directly to a computer.
Clicking the "Connect" button will automatically launch the Splashtop Business client app, log in with your Splashtop ID, and initiate connection to the appropriate computer.
The Splashtop Business client app may prompt you to enter your password, if the app has not been launched before or a password has not been saved.
You can download the latest Splashtop Business client apps from www.splashtop.com/downloads#b.
What are those browser warnings?
In order for the browser to launch the Splashtop Business client app, it may need to get permission from you. Just allow it to proceed.
You can also check the box to prevent these warnings from being shown in the future.
Other notes
If a session is initiated via the web console, when the session terminates, the Splashtop Business client app will close itself and return you to the web console
On Windows, you can launch multiple simultaneous sessions, just by clicking "Connect" for multiple computers in the web console. Each session will open in its own instance of the Splashtop Business client app.
Unfortunately, above does not work on Mac due to some OS restrictions. Launching a session from the web console on a Mac will only switch to the already-running instance of the Splashtop Business client app.
However, it is possible to open multiple instances of the Mac client manually via the following command in Terminal (not using the web console):
open -n -a "Splashtop Business"