The file you download is League of Legends NA.dmg, and it is a disk image; This is one of the standard ways to distribute a mac application. Double clicking a DMG file "mounts" it much the same as if you inserted a DVD or USB stick, and it opens a Finder window that shows the contents of that disk image.
Generally you'll see the application and a shortcut to your applications folder in this window, often with a graphical background. The idea is to help you copy the application from the disk image over to your Applications folder. Once you've copied it over, you want to press cmd+e to eject the disk image. From there, you just need to navigate to your Applications folder in finder, and double click the new application to run it.