A roux thickens a sauce because flour starch, once dispersed through hot fat, can absorb water and swell without clumping. Fat coats individual granules and separates them before liquid is added — the same reason a dry spoonful of flour dropped into milk seizes, while flour cooked in butter does not.
When warm milk or stock meets the roux, water penetrates the swollen granules and amylose molecules leach into the surrounding liquid. Gentle heat keeps the mixture moving so hydration is even and the sauce stays smooth. Push the temperature too hard and the starch can break down or the milk can scorch at the base of the pan.
As the sauce cools slightly, amylose chains begin to set and the liquid gains body. That setting is reversible with heat, which is why a béchamel can be loosened and tightened again on the hob — provided the roux was cooked enough to lose its raw flour taste but not so long that the starch loses thickening power.
The practical lesson is structural rather than chemical trivia: dispersing flour in fat first, adding liquid gradually, and holding gentle heat gives you control over thickness without lumps. A straight-sided saucepan concentrates the liaison; a whisk keeps the starch evenly suspended while the sauce thickens.
White roux, blond roux and brown roux differ chiefly in cooking time and flavour, not in the underlying mechanism — longer cooking develops nuttiness while slightly reducing thickening power. Naming which roux you have made helps when a sauce tastes of raw flour or when it will no longer set as expected.