Distilled water to clean greasy stains: Why does that work?
There may be two different factors at play here.

First, it isn't necessarily the water that is doing most of the cleaning. I presume the water is used to wet a cloth and the cloth then wipes the surface. So the cloth may then physically remove surface contaminants. The role of the water may be just to improve the cloth's surface contact hence improving its ability to physically remove the contaminants.

Secondly, distilled water might do a better job than tap water as it has fewer dissolved solids which would leave other stains on the surface when the water evaporates.

However, the only way to be sure is to do a comparative experiment using different cleaning fluids. You might try tap water, distilled water and distilled water with a surfactant (e.g. a liquid soap). You might find that the best result uses the surfactant but that pure distilled water does a less good, but good enough job. Science isn't about single isolated experiments: we usually need to compare alternatives to know what is best.