In the email that Amazon sends you, there is a section that says, "How to redeem this item." Just below that (not the next line, but within the next 3 -- on the email they sent me anyway), there is a line that says, "Step 1." That line contains some text in blue that says, "Digital games and software". When you click on this link, it takes you to Amazon. Amazon may make you log in, but then it will take you to the place where you can download the software from.
After downloading the software, open This PC or File Explorer. For clarity, let's say you used This PC. In This PC, navigate to the place where you downloaded the software to. Once you locate the software, double-click on the file you want to run.
After installing the software, when you start up TurboTax, somewhere in the process (I don't remember exactly where), it will ask you for the license.
In my case, I had purchased Deluxe from Amazon, installed it, started using it, and later discovered that I needed Premier. At the time, I didn't know that you could upgrade from within TurboTax... So I purchased a new full version of Premier from Amazon, and I was trying to manually install the new version. I uninstalled Deluxe, and then installed Premier. When I started up TurboTax, it kept identifying as Deluxe. After some uninstall/reinstall iterations, I discovered, on the Menu bar, under Help, an option to Activate product, and it had my new version. Going through the Activate dialog, it had me supply the new license. After that, it had me restart TurboTax, and then TurboTax identified as the new version I had purchased.
