Airconsole is meant to be played using a smartphone, it doesn't even let you connect with a PC. To connect with a PC, right click the page on airconsole.com and press "inspect element". A console will appear. Click the smartphone/tablet emulator button top right on the console page. It is a bit hard to find. When it is clicked, your screen will resize. Hit ctrl+r or F5 to refresh, then try joining a game.
In most games you will have to control using your mouse, and the experience will be terrible (since its meant for phones...) In supported games, this script will bind the onscreen buttons to keyboard controls. These are mainly WASD, arrow keys and SPACE. Here is a list of supported games and the keys they use:
Main menu screen (WASD)
Tower of Babel (S/Space)
Racing Wars (WASD, space?)
Battle snakes (WASD)
Arrow keys should also work as a replacement for WASD if you preffer that.