You need to right-click on the download link and choose Save link as....
This is important especially for Firefox users, as Thunderbird and Firefox both use the XPI extension for their plug-ins. If you click on the download link, Firefox will try to install Enigmail as a Firefox plug-in, which will not work. Use Save link as... instead and save the XPI file on your machine.
Start Thunderbird. In the menu bar of the main window, select Tools →Add-ons to bring up the Thunderbird Add-ons Manager window.
Select Extensions, then click the Install button and tell Thunderbird where you saved the Enigmail XPI file. Another window will pop open, warning that you're about to install a plug-in. Confirm your decision.
You will then need to restart Thunderbird. Once Thunderbird has restarted, Enigmail will be ready to go.