Crickets sing by stridulation: a scraper on one forewing is drawn across a file of teeth on the other, and a resonant patch of wing radiates the sound. The rate at which the wings can be cycled depends on muscle contraction speed, and insect muscle has no internal thermostat, so it follows the temperature of the surrounding air.
Amos Dolbear published the relationship in 1897, and the rule that carries his name is usually stated for the snowy tree cricket: count chirps in fourteen seconds and add forty to obtain degrees Fahrenheit. Applied to other species the constants must change, because each has its own baseline chirp rate and song structure. The relationship is close to linear over the moderate temperatures a cricket is active in, and degrades at the extremes.
The common error is treating it as a general property of crickets. It is species-specific, and the field cricket's continuous chirping is not the same signal as the snowy tree cricket's regular pulses, so Dolbear's constants do not transfer.