I looked around on the buffalotech.com site but couldn't find a product listing for the model you listed (HS-DH0.0TGL/R5); there are several similar models listed (HS-DH1.0TGL/R5, -DH2.0, etc). You want to double-check that you've downloaded the correct firmware for your model.
If the firmware is downloaded as a .ZIP file, you need to extract its contents. There should be an .EXE file named "BuffaloFirmwareTeraStation112-104HD-HTGL FWUpdate.exe" or similar inside the .ZIP.
Make sure you're doing this from a wired connection (do NOT update the firmware over wireless). You may also need to disable your antivirus and firewall.
Once you've booted the Terastation and seen the data copied, run the firmware update .EXE you extracted. (Double-clicking the .EXE should be sufficient.) That program searches for the NAS on your network, checks that the model of the NAS matches the model of the firmware, and updates the firmware.
If you've gotten the firmware for the wrong model, the updater will take no action.