The extension is optional. It marks Nexus pages with what your profile already has and sends Mod Manager Download clicks to Mortar. Settings › Downloads has a Download the extension button for mortar-browser-extension.zip from its releases, and says The browser extension is too old for this Mortar (or too new) when versions drift.

Chrome, Edge, other Chromium: install it from the Chrome Web Store, which keeps it updated. To try the zip instead: unzip, open chrome://extensions, turn on Developer mode, choose Load unpacked and pick the folder.
Firefox: open mortar-browser-extension.xpi from the latest release in Firefox, or drop it on about:addons; Mozilla signs it, so it stays. Without an .xpi, use the zip: about:debugging, This Firefox, Load Temporary Add-on, pick manifest.json (removed when Firefox restarts).