Super simple answer, so apologies if it sounds condescending or I've glossed over aspects entirely.
Libs like vue & react are essentially ui renderers, they create the html representation of app state, and handle all the DOM event binding and whatnot for you.
You will likely have some kind of client state management to take care of telling your ui to update, but essentially you will push json into that 'store' and that will then be used by your framework of choice to populate the page with content.
How that json gets there is largely upto you, and will be influenced by what you use to manage state, however you will need to handle getting data from the db server-side. Php or rails are popular, with reams of great (and some not so great) resources out there, but realistically you could use whatever you wanted.
Node.js is nice and could be useful if you want to focus on 1 language initially.