The Tweaks for Twitter browser extension is bundled inside the Mac app. You only need to enable the extension once in your browser, and then it will automatically update along with the Mac App Store app.
The instructions below are for Google Chrome. You would follow similar steps to enable Tweaks for Twitter in another Chromium browser such as Brave, Microsoft Edge, Opera, or Vivaldi. The user interface in other browsers may look slightly different, but otherwise the process is the same.
Purchase and install Tweaks for Twitter in the Mac App Store.
Open Google Chrome.
In Chrome, open the "Window" menu in the main menu bar and select "Extensions" from the menu.
In the Extensions window, enable "Developer mode" in the upper right corner.
Open the Tweaks for Twitter app.
In the Tweaks for Twitter app, select "Enable in Chrome or Chromium Browser".
Drag the widget out of the "Enable in Chrome or Chromium Browser" window and into Chrome.
Drop the widget into the Chrome Extensions window. Tweaks for Twitter is now installed in Google Chrome.
If you wish, you can disable "Developer mode" after enabling Tweaks for Twitter. The extension will continue to function and receive software updates.
If you see an Errors button, don't worry, this is actually just a warning about the deprecated Chrome extension manifest version 2.
Tweaks for Twitter version 12.0 adopted the newer Chrome manifest version 3. If you still have the old manifest version 2 Tweaks for Twitter extension installed, you can update by pressing the Remove button as shown below, and then reinstalling the Chrome extension using the instructions above. Afterwards, you may need to set your Tweaks preferences again.