One of our engineers, Jordan Walke, was frustrated with the existing tools for building web UIs while working on some of our tools for advertisers on Facebook. We were using a custom-built library similar to Backbone where you declared the initial appearance of a view completely separately from how to update it. Jordan had the idea to automatically apply updates by rerendering each view and calculating a difference.
Apparently we were pretty skeptical of the idea and discouraged him from working on it. He kept building it anyway and eventually proved to everyone that it could work and could be fast enough. I'm not sure how the team grew from there but we started using it in more places like the Like/Comment/Share interface on every piece of content and the Instagram website, which was built from scratch in 2012 and 2013. Eventually React grew to be used in more projects and was released as open source.