There is no magic in these widgets. People have to add certain snippets, mostly a JavaScript file from some CDN and some basic HTML tags where these widgets shall appear, and everything else is done by the JavaScript.

In former days, people used iframes, this was even easier because the iframe could basically show any website with its own styling, scripting, etc.

If you want to get hands dirty, then skip any framework, learn modern ECMAscript, vanilla Dom and try out to modify a well known element in the target page from your script.