Why chrome developer tool does not show WhatsApp web app requests?

You have to open the Google Chrome inspector before the WebSocket connection is opened. Then as @Hernan has pointed out, you go to network -> websockets, select the connection with 101 status and in the frames tab you will see the messages. But you have to switch from "Frames" to "Headers" tab and come back again in order to Chrome refresh the frames.

If you send messages with the Frame tab opened, it does not update automatically. Or if you open the Chrome inspector after the WebSocket connection has been established, you won´t see the connection.