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TLDR; The code is compiled and download in the client side.
Vue is a library , which in very simple terms means you can access re-usable code that has lots of useful functionality and use it to build your application much faster and in a cleaner way.
You can think of it as an external resource where you are taking ready-made code in order to build your app, so no real magic happens here, just a lot behind the scenes that saves you lots of effort.
Now when it comes to compiling, Vue CLI comes ready with Webpack, Router, and more, so it automatically handles Routing for you, compiling your code, making your bundle size smaller, uglifying it, code-splitting different parts and more.
If you want your code to be compiled on the server and then served to the client (as it’s called Server Side Rendering) you can use Frameworks such as Nuxt.JS that works with Vue in order to achieve that.