You will get to keep the logic and structure of the components, how ever you would have to rewrite the UI used in them.
The approach is different from other cross-platform frameworks out there (like ionic with phonegap or xamarin) which is code once, run everywhere with a single code base.
The approach of React Native is, learn once, write everywhere. This means that the same team of engineers who built the iOS version will be able to understand enough to build the Android version however they will still require some knowledge of it to do it.
With React Native will end up having two separate projects, which is alright because they are for two different platforms, but their internal structure will be very similar.
You can see what Im talking about here: https://rnplay.org/apps/sXriww This is a React Native component built for iOS and Android. They are two different files (two code bases) with different UI templates but the structure in them is the same.