This download looks like an addon. You need to extract the downloaded zip and put the addon folder in your Garry's Mod addons folder. You should end up with a directory structure similar to below:

• steamapps
  • common
    • garrysmod
      • garrysmod
        • addons
          • Weather Mod <-- Addon folder!
            • e.g.
            • lua
            • maps
            • materials
            • models
            • etc

• Weather Mod/
  • info.txt
  • lua/
    • autorun/
      • client/
        • cl_weather.lua
      • server/
        • sv_weather.lua
      • sh_weather.lua
    • effects/
      • rain/
        • init.lua

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