flatMap() is a specific function that you already understand. It is the most common instance of a general fn pattern that is used for the specific data type Array.
The flatMap equivalent function when used on the data type Promise is called then().
The special function you are asking to understand ">>" is simply another instance of then(). This one still waits for the promise to complete - the only difference is that the lambda you pass to then() doesn't have access to the result of the previous promise.
If you're using >>/flatMap on a Parser datatype, instead of running the lambda after first promise has completed successfully, it "runs" it after the first parser has completed successfully.
flatMap and (all of its other data type specific names) are the essence of a Monad.