A condensate splitter is simply a distillation tower. To understand how a distillation tower works, you really need a course in mass transfer. Very simply, the light materials go out the top of the tower and heavy materials go out the bottoms. There are always some lights in the heavy stream and vice versa, how much depends how you design the tower and how you operate it.
As to why you have a condensate splitter, that depends on the application. It might be that they have a vapor pressure specification on the natural gasoline and need to remove some of the light material to meet that spec.
It can also depend on where the material is being sent to. Natural gasolines might be a feedstock for a reformer to make gasoline. Reformers work best with heavier materials, C5 and lighter materials aren't reformed and just take up space in the system which could be used more benefically by heavy material. Thus, you might want to remove the light materials from the natural gasoline to increase total gasoline production from the refinery.
That's just one example, there could be hundreds of reasons why a plant/refinery would want to have a splitter. But since it costs money to install it and operate it, there was a reason at one time and it would be driven by the process requirements and specifications.