Why do milk or other fats extinguish chili heat more effectively than plain water?

They dissolve the capsaicinoid oils.

Capsaicinoids, the chemical family responsible for pungency, possess oleaginous or oily properties. Because capsaicin is oil-soluble, substances containing fat or lipids—such as milk, yogurt, or creamy sauces—are highly effective at dissolving and washing away the compound from the sensitive receptors in the mouth. Conversely, drinking plain water is largely ineffective because water is polar and does not mix well with the non-polar capsaicinoid oils. Attempting to wash away the heat with water often results in spreading the oils across a wider surface area of the mouth, potentially intensifying the localized burning sensation until the oil can be physically dissolved or rinsed away by something fatty.