So does it use PhoneGap or something like it?

No, because PhoneGap uses the DOM.

No, it skips the DOM entirely and works with native elements (iOS views) directly.

This is where I get confused. How is it compiling to these elements?

One possibility: You have a JS interpreter running your React code. You generate a virtual DOM and hand it off to React. React sends the virtual DOM through a message bus to native code. So far, nothing crazy here, this is how frameworks like cordova work. That native code dynamically translates the virtual DOM into native UI elements.

That's the million dollar question and tomorrow we might get an answer.

Oh snap. This is HUGE