How do you use React.js for SEO?

I'm pretty sure anything you've seen promoting React as being good for SEO has to do with being able to render the requested page on the server, before sending it to the client. So it will be indexed just like any other static page, as far as search engines are concerned.

Server rendering made possible via ReactDOMServer.renderToString. The visitor will receive the already rendered page of markup, which the React application will detect once it has downloaded and run. Instead of replacing the content when ReactDOM.render is called, it will just add the event bindings. For the rest of the visit, the React application will take over and further pages will be rendered on the client.

If you are interested in learning more about this, I suggest searching for "Universal JavaScript" or "Universal React" (formerly known as "isomorphic react"), as this is becoming the term for JavaScript applications that use a single code base to render on both the server and client.

As the other responder said, what you are looking for is an Isomorphic approach. This allows the page to come from the server with the rendered content that will be parsed by search engines. As another commenter mentioned, this might make it seem like you are stuck using node.js as your server-side language. While it is true that have javascript run on the server is needed to make this work, you do not have to do everything in node. For example, this article discusses how to achieve an isomorphic page using Scala and react:

Isomorphic Web Design with React and Scala

That article also outlines the UX and SEO benefits of this sort of isomorphic approach.