Right-click the downloaded file SDLTrados2007Suite_PRO_863.exe and choose 7-Zip -> Extract to "SDLTrados2007Suite_PRO_863\".
Open the newly created SDLTrados2007Suite_PRO_863 folder and then open the SupportFiles folder within it.
Double-click SDLTPRO2007SP3.EXE and follow the on-screen instructions to install it.
When the installation has finished, go to Control Panel -> Programs -> Programs and Features.
Find and click the latest version of Studio that you use, for example SDL Trados Studio 2019 (do not click the suite uninstaller e.g. SDL Trados 2019 - Remove suite of products)
Click Repair above the list of programs and wait for Studio to complete the repair process.
Open SDL MultiTerm Extract. You should now be able to add all of the supported files to your project.
Note: If Trados 2007 suite is already installed, please repair it from Control Panel.
Root Cause
The core components of SDL Trados 2007 are not installed. This is required for full functionality of SDL MultiTerm Extract.