This update speaks to the theories on enzymes, offered in other answers.

It is useful to refer to ectotherms (an organism in which internal physiological sources of heat are of negligible importance in controlling body temperature, which permits them to operate at very economical metabolic rates) as poikilotherms when applicable.

"Poikilothermic animals include types of vertebrate animals, specifically some fish, amphibians, and reptiles, as well as a large number of invertebrate animals. The naked mole-rat is the only mammal that is currently thought to be poikilothermic.".

"Poikilotherm animals must be able to function over a wider range of temperatures than homeotherms. The speed of most chemical reactions vary with temperature, and in order to function poikilotherms may have four to ten enzyme systems that operate at different temperatures for an important chemical reaction. As a result, poikilotherms often have larger, more complex genomes than homeotherms in the same ecological niche. Frogs are a notable example of this effect, though their complex development is also an important factor in their large genome.

Because their metabolism is variable and generally below that of homeothermic animals, sustained high-energy activities like powered flight in large animals or maintaining a large brain is generally beyond poikilotherm animals. The metabolism of poikilotherms favors strategies such as sit-and-wait hunting over chasing prey for larger animals with high movement cost. As they do not use their metabolisms to heat or cool themselves, total energy requirement over time is low. For the same body weight, poikilotherms need only 5 to 10% of the energy of homeotherms.".

This update speaks to the theories on enzymes (solitary from metabolism), offered in other answers.

It's a balance between preventing infection and not requiring excessive food intake.

"The number of fungal species that can thrive and therefore infect an animal declines by 6 percent for every 1° C rise in temperature. This means that tens of thousands of fungal species infect reptiles, amphibians and other cold-blooded animals, but only a few hundred harm mammals.".

It's a balance between preventing infection and not requiring excessive food intake. It is has also been claimed that this is the temperature at which the most essential enzymes are most reactive, without becoming damaged by excess temperature.

"The number of fungal species that can thrive and therefore infect an animal declines by 6 percent for every 1° C rise in temperature. This means that tens of thousands of fungal species infect reptiles, amphibians and other cold-blooded animals, but only a few hundred harm mammals.".

This update speaks to the theories on enzymes, offered in other answers.

Articles and theories that are not discredited:

• Enzyme assays by Hans Bisswanger:

That article is mostly concerned with the temperature used to conduct assays of enzymes.

"The essential requirements for enzyme assays are described and frequently occurring errors and pitfalls as well as their avoidance are discussed. The main factors, which must be considered for assaying enzymes, are temperature, pH, ionic strength and the proper concentrations of the essential components like substrates and enzymes. Standardization of these parameters would be desirable, but the diversity of the features of different enzymes prevents unification of assay conditions. Nevertheless, many enzymes, especially those from mammalian sources, possess a pH optimum near the physiological pH of 7.5, and the body temperature of about 37 °C can serve as assay temperature, although because of experimental reasons frequently 25 °C is preferred. But in many cases the particular features of the individual enzyme dictate special assay conditions, which can deviate considerably from recommended conditions.".

...

"Enzymes display their highest activity at their respective optimum conditions, deviations from the optimum cause a reduction of the activity, depending on the degree of the deviation. Moderate deviations produce only small activity decreases which can be tolerated (Figure 1), and so the physiological conditions prevailing in the cell may be taken as standards for at least of the mammalian enzymes. However, assay procedures are usually adapted directly to the features of the individual enzyme and not to obey general standards. Enzymes are sensitive substances present in small amounts and their activity in the cell can often be detected only at their optimum conditions.".

• Royal Society of Chemistry - "Enzymes - Function and structure":

"As the temperature rises, reacting molecules have more and more kinetic energy. This increases the chances of a successful collision and so the rate increases. There is a certain temperature at which an enzyme's catalytic activity is at its greatest (see graph). This optimal temperature is usually around human body temperature (37.5 °C) for the enzymes in human cells.

Above this temperature the enzyme structure begins to break down (denature) since at higher temperatures intra- and intermolecular bonds are broken as the enzyme molecules gain even more kinetic energy.".

Note: Wikipedia: "Wilson's temperature syndrome":

"The American Thyroid Association (ATA) describes Wilson's syndrome as at odds with established knowledge of thyroid function. The ATA described the diagnostic criteria for Wilson's syndrome as imprecise and non-specific, and raised concern that the proposed treatments were potentially harmful. Florida State Medical Board members described Wilson's syndrome as a "phony syndrome" and a scam during disciplinary action against Wilson."

...

"The ATA stated in 2005 that a "thorough review of the biomedical literature has found no scientific evidence supporting the existence of 'Wilson's Temperature Syndrome'." The statement added that the mean temperature of normal persons in the AM on waking is 97.5 °F, not 98.5 °F, and that many of the symptoms described by Wilson are nonspecific and typical of depression, anxiety, and psychological and social stress. It also notes that a similar set of symptoms occurs in the alternative diagnoses of neurasthenia, chronic fatigue syndrome, fibromyalgia, multiple chemical sensitivity, chronic Epstein-Barr virus syndrome, and chronic candidiasis. Finally, the Association notes that chronic supplementation with triiodothyronine (T3) is particularly difficult and problematic, since various tissues set their own cellular levels of this hormone by making it individually from thyroxine, and supplementation of T3 may overwhelm this normal regulatory mechanism in some of these tissue.".

Reference: Wilson's Temperature Syndrome

• "Metabolism Dependent on Enzymes"

• "Enzyme Function Dependent On Temperature":

"... when an enzyme-dependent chemical reaction is monitored for how well it takes place at various temperatures, the lower the temperature, the slower the chemical reaction. As the temperature is increased, the reaction rate will go faster with each increase until it reaches its optimum reaction rate. If the temperature is increased too much, the rate of reaction will diminish due to denaturing or change in shape of the enzyme.".

The treatment suggested by Wilson is what was discredited, the fact that enzyme reactions proceed differently at different temperatures is still supported. Reference to his article was included due to the partial overlap with established knowledge.