After using Hybrid Connection, i wanted to figure out how it works. Here are the notes:
When setting up the "on-premise" software, a Windows Service "Azure Hybrid Connection Manager Service" is deployed on the machine which launches a listener application "Microsoft.HybridConnectionManager.Listener.exe" present at "C:\Program Files\Microsoft\HybridConnectionManager".
Process explorer shows that the executable is launched by the service to open a persistent (?) connection of some sort with the azure website.
You can verify it by watching the TCP/IP connections used by the "Microsoft.HybridConnectionManager.Listener.exe" in process explorer.
In my application, I accessed SQL Server hosted on my local machine. As shown in the above screenshot, communication address uses ":ms-sql-s" to indicate that.
Interestingly the connection with remote host is closed after ~60 seconds if there is no more traffic between the two machines.