Where did this myth come from?
Martial arts instructors are frequently tasked with running a sport workout without having any formal training (that is, outside of their own dojo) to do so. This means that many of these instructors--my past self included--ran classes that were particularly counterproductive from a workout-design perspective.
One of the most common ways this manifests itself is with overly intensive "warm-ups". Really, these are conditioning workouts in their own right, though they are generally not structured in a progressive fashion to realize training goals over time. This is common in any widespread exercise method. For instance, this criticism of CrossFit notes that there's a distinct tendency to emphasize "beat-down" workouts instead of rationally designed progressions towards training goals. It's easy to run students through a grueling set of exercises and make them tired.
It requires time, knowledge, forethought, and effort to run a structured workout or design a training program correctly. That's why the ex-post-facto "training when you're tired will make you learn the technique better" explanation (and it's cousin the just-so story of "training when you're tired means you'll remember the technique when you're tired") is so popular.
Tom Kurz' Science of Sports Training page 267 (under Technical Exercises in a Workout) notes:
If, at the initial stages of learning a technique, athletes are allowed to get tired, their fatigue will alter the technique and incorrect technique will be learned, perhaps permanently.
(One of Kurz' articles on this topic, entitled Examples of Good and Bad Workouts, used to be available on stadion.com but is currently unavailable.)
When should this method be used?
From the same book but on the next page, it is noted that techniques the student knows and can already perform well should occasionally be trained under some degree of fatigue:
One [technical] workout can be done at the end of the [weekly] microcycle, when the athlete is tired [from previous workouts in the week], to learn how to use the technique in adverse conditions. In this workout only well-mastered techniques should be practiced.
This is contrary to the idea that one should learn the technique when tired, which interferes with learning. It only applies to techniques the athlete is proficient in, and now needs to practice under more challenging conditions.
What's a better approach?
For learning a new technique, a better approach is to run a structured workout that progresses from warm-up to directed warm-up to sport-specific movements and culminates (while the athlete is still fresh) in learning one technique. Learning multiple techniques causes interference and should be avoided.