First you must purchase the game by registering account into frontier store, you'll find Elite Dangerous category in there.
Then you purchase the game. After the purchasing process is done you'll get the game title to show up in My Products page.
Click and download the elite client installer exe from My Products page. This is the installer that is used to, well install the game. The installer is not the game, you use the installer to download the game from internet.
If you have trouble finding the installer link, here is the direct link to hosting.zaonce.net/elite/Client-Installer.exe file.
Run the installer and point to some proper directory (not c:\program files\ oh no) and wait, depending on how busy frontier servers are and what is your internet connection speed (and location), it will take a while for the download and install to complete.
Once the game has been installed you can Login with your email/password you created in frontier store.
You can read more about using Elite Dangerous launcher in multiple accounts page.