Navigate to the directory:
..\Where You Installed Steam\steamapps\YOURSTEAMIDHERE\garrysmod\garrysmod\addons
Please note that you need to replace YOURSTEAMIDHERE with your actual Steam ID.
For example, I am running Windows 7, my Steam ID is sample123, and I installed Steam to:
C:\Program Files (x86)\Steam\
So I would navigate to:
C:\Program Files (x86)\Steam\steamapps\sample123\garrysmod\garrysmod\addons
Go to your zip file and drag the folder "HenrysAnimationTools" to the addons folder you've just opened.
Congratulations, hopefully you've installed HAT correctly into Garry's Mod. The next step is setting up HAT.