Batch can trigger with a number of analytics-oriented events that might be of interest in external analytic tools. An event dispatcher is the code that listens to those events, and dispatches them to the solution of your choice.
Event dispatchers are implemented using native code on both iOS and Android as they are a low level feature of the SDK.
Batch provide you with premade dispatchers for specific use cases (aka ready-to-go dispatcher). They will automatically track events into your analytics solution with a predetermined structure.
As React Native makes it easy to tweak the native projects, please follow the native event dispatcher documentation for each platform:
Once you add the native dependency and register the dispatchers (when needed by the platform), any SDK interaction triggered from React Native will automatically be tracked in the event dispachers.