The reason why you should not drink water after eating spicy foods is because the water can make the spiciness worse.
The capsaicin which is the active compound in peppers is what is responsible for the spiciness, is not soluble in water.
Capsaicin is a hydrophobic molecule that hates water and binds to pain receptors in your mouth which causes the burning sensation from spicy foods.
When you drink water after eating spicy foods, it does not dissolve or neutralize the capsaicin and instead water can spread the capsaicin and make the spiciness worse.
Instead of drinking water after eating spicy foods to cool down the heat you should drink liquids that can dissolve the capsaicin.
Drinks and foods that can dissolve the capsaicin are cold drinks other than water, bread, yogurt and milk.
The starch in bread can absorb the capsaicin and reduce the burning sensation.
Yogurt, similar to milk contains fat and proteins that can also help with the spiciness.
And milk has fat and proteins which can bind to the capsaicin and neutralize the effect of the capsaicin.