Drag the Electrum.app icon into your Applications folder. This appears to the left of the window under Favorites. Open the Applications folder and double-click the Electrum icon.
Quite often, it is this step that gets people... you need to drag the Electrum.app into your Applications folder to get it to "install".
If you simply open (double-click) the downloaded ".dmg" file and then double-click the Electrum icon, it will run Electrum... but when you restart your computer, the .dmg file won't be "mounted" (open) anymore... and you won't find the Electrum app. You would need to manually open the ".dmg" file again, and then open Electrum again... and do that every time you restarted.
Better to drag the .app into your Applications to install it