I don't know how and why you found this page but in my case, my firefox has trouble auto-installing the plugin. To be precise, it's experiencing strange network problems while downloading (connection to a Google domain like 'r1---sn-bvn0o-fo1l.gvt1.com' always times out when I use a VPN).

So, I have to do it manually. At first I searched the internet and found someone experiencing the same problem. I took a look at the guy's solution (https://www.reddit.com/r/firefox/comments/9s07bh/tips_how_to_manually_install_widevine_plugin_to/) and I summarized it a little bit.

FIRST,

Enable 'Play DRM' in the browser's settings page. Go to https://github.com/mozilla/gecko-dev/blob/master/toolkit/content/gmp-sources/widevinecdm.json to find the latest release of the plugin that fits in your browser. The whole shitass JSON file will be shown. You need to find the right URL to download it. For example, you are using 64-bit Windows system, so you need to find a text with "WINNT" and "x86_64" included and download the file using the URL next to it. This matches what you need:

"WINNT_x86_64-msvc": {
  "fileUrl": "https://redirector.gvt1.com/edgedl/widevine-cdm/4.10.2391.0-win-x64.zip",
  "filesize": 6537814,
  "hashValue": "81b2329d38a9370afc490db805a05c4b506b113ebd00f4e488bca97fd96267d92cb477e3a635880464ca66ed32f448e46ad3645f6af072547b5f09100db2bf74"
},

In my case, I'm using 64-bit KUbuntu so I need to find the text with "Linux" and "x86_64" included and use the related URL:

"Linux_x86_64-gcc3": {
  "fileUrl": "https://redirector.gvt1.com/edgedl/widevine-cdm/4.10.2391.0-linux-x64.zip",
  "filesize": 6328796,
  "hashValue": "8ce16faae96274e1f5ec63f6f543fa33ab3b7d469e59fac2d8b45cb27d3c95820cf80cd362d6e972a1c3c27e5c1b28c018fbdc2bb7df50f095391a646e277a99"
},

The way to download a file is out of the scope of this article.

SECOND,

go to your browser's 'about:support' page. Find the 'Profile Directory' and click the 'Open directory' button next to the text. Find a directory called 'gmp-widevinecdm'. If this directory not exists, make one. Go to this directory and make a directory named by the plugin's version. Open the zip file and extract all contents into this directory.

THIRD,

go to your browser's 'about:config' page and set the keys and values like this: Type of media.gmp-widevinecdm.lastUpdate is Number. Type of ...autoupdate, ...visible and ...enabled is Boolean. The rest is all String. Remember to adjust the value of ...abi to suit your situation. Now go to the plugins page and you'll see the plugin is successfully installed.

Note:

The wrong configuration in about:config seems to cause the plugin to disappear after a browser restart.