Yep you need a webserver. For a quick approach try installing http-server which spins up a simple one-off host. Ultimately you want to serve your app though either standalone via e.g. nginx or as static part of a backend
If you are using webpack, the npm run build will build everything in dist/ folder, it has an index.html file and all js/css bundles. You would need to copy all those files to a http server root folder.
dist/index.html to root/index.html
dist/appxxx.js to root/appxxx.js
If you are using cli built-in bundler, the npm run build will build index.html file in your project folder, and all js bundles in dist/ folder. You would need to copy all those files to http server root folder following the folder structure.
index.html to root/index.html
dist/xxx-bundle.js to root/dist/xxx-bundle.js
Before deploy to production, you can use a local server (http-server as zewa666 suggested) to test it out.
npm i -g http-server
Then run http-server inside dist/ folder for webpack, or run inside just project folder for cli built-in bundler.