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, 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.