React Native has the same principle as for React. Developers can write code in a declaritive way as a number of components. For ReactJS on the web these components make up a shadow Document Object Model (DOM) which react will render to the screen in an efficient way, changing only what needs to be updated.
The translation of the shadow object model for React Native is more complicated than for React on the web. The styles do not directly translate to enable flexbox layout for example. In order to do the UI translation needed for mobile Facebook have developed a library called Yoga which does this job.
New architecture is being developed to improve on the current system. In particular the communication over the bridge in JSON can be a bottle-neck and this is going to be replaced.