When I googled about React.js what I got is: React.js is a Framework that is used to create user interfaces. If a particular part of the website is frequently updated that means
Ajax is used to refresh a web page without having to reload it : it sends a request to the server, but typically the response is processed by the javascript that displays dynamically a new element on the browser without having to reload the entire page.
React is a javascript library that dynamically update the page with inferface components. The components are calculated either by javascript interactions or by an ajax request that go through the server. So ReactJS can also use Ajax requests to update the page.
Mustache and Handlebars are a bit different from ReactJS as the main goal is to transform a template in a component that will be displayed in a page. It can also use Ajax to get data (for getting templates or json datas).