It's an overused axiom, because it's so often misapplied.
While the the rule, called the "rule of Arrhenius", predicts a doubling of a chemical reaction rate upon every 10°(centigrade) increase in temperature, it does not follow that there would be a corresponding measured increase in the activity or process of cleaning.
This is mainly due to the fact that cleaning, as we know it, is principally a physical process, not a chemical reaction, per se. Yes, heat does aid cleaning, just not at any predictable increase as it relates to the increase in heat.
As well, the "Rule", is more of an observation as it pertains to a subset chemical chain reactions which are heat dependent. A few reactions are in fact not heat dependent, and are hardly effected by an increase in heat, while others can be accelerated exponentially by a slight increase in heat. So, the "rule" becomes a "rule of thumb", more than anything else.
However, heat is an excellent accelerator for physical processes too. As we add heat energy, we speed up molecular movement. Physical reactions are often aided by molecular movement to overcome forces of repulsion or boundary layering that keeps solvents from actively dissolving solutes, or in this case soils. Heat energy also aids surfactants in their ability to surround oil droplets, and heat combined with physical agitation helps form smaller oil droplet sizes for the surfactants to form surrounding structures, called "micelles". it is these surrounding structures, or micelles that permit surfactant(detergents) to "dissolve" and suspend oil into water to make it able to be flushed from the fibers with extraction or rinsing.
However, every surfactant has a limitation of the effective temperature, called its "cloud point". Simply put, the cloud point is the upper temperature extreme, that when reached, the surfactant's structure falls apart. The surfactant itself is no longer able to be dissolved in water, and it "clouds" the solution. As well, the surfactant is unable to form micelles to surround fat/oil droplets, and the process of detergency is abated. In most cases, returning the solution to below the cloud point restores the activity. It's not as though overheating a surfactant a little in the heater, or in the solution line destroys anything, so long as it remains dispersed and cooled below the cloud point as it is applied to the surface being cleaned.