Okay, this confuses me. As it gets colder doesn't it mean more possibility of icing? And is the -40 degrees SAT or TAT?
That -40 is SAT and in degrees Celsius.
The reason that anti-ice is generally no longer needed below that temperature is because at that point, the air is so dry that there isn't enough moisture left to form frost or ice that will stick to the airframe or engine parts.
Normally, it's engine anti-ice, rather than wing anti-ice, that is still on when you get to the point in the climb that the temperature drops below -40. At least in the 737, the engine anti-ice is kept on continuously while the temperature is below +10 TAT and above -40 SAT and you're in IMC or other visible moisture (i.e. rain, snow, etc), while wing anti-ice is only used "as needed" to remove ice buildups from the wings.
Also of note, the "below -40 is too cold" logic only exists in climb and cruise, not in a descent. The rationale there being, you don't want to miss the transition from the air is no longer "too cold for icing" as you descend, and find out that your motors ARE now accumulating ice, when they start to act sick... that makes your day entirely too interesting!