In the "Download Now" section of the sidebar on the right, click the version you want to download (OmniDiskSweeper or OmniDiskSweeper Pro).

Wait until the download is complete. If your web browser and system are properly configured, Disk Copy or Finder will open and mount the downloaded disk image. If this does not happen, locate in Finder the .dmg file that you downloaded and double-click it to open the file. (Double-check to be sure that you've quit all running copies of OmniDiskSweeper before you continue any further...)

When the disk image mounts, a new Finder window will open that contains the icon for your new OmniDiskSweeper application. Simply drag this icon to your Applications folder (or wherever you would like to keep it) to install. This copies the application from the virtual disk image onto your hard disk so the application is there for you to use whenever you like.

Once OmniDiskSweeper is installed you can safely 'eject' the virtual disk that it came on and also dispose of the .dmg file that you downloaded if you choose. Double-click the OmniDiskSweeper icon in the location you installed it in order to begin using the application. You may wish to drag the icon to the Dock for easy access.