The claim is that we should be drinking water routinely to the point that we never feel thirsty, and that to not be doing so is to our detriment, at least enough for the notion to be popular amongst hikers, gym teachers, athletes and the like.
While I understand that from a medical point of view at the onset of thirst we are "mildly dehydrated" (thirst wouldn't make much sense if nothing was going on), the claim is that the sensation is indicative of an already-made mistake in behavior, and that drinking water now will be of significantly less benefit than if we had beforehand, preempting thirst.
Here is an example of one side of the claim:
If you wait until you are thirsty, your body is already dehydrated. It takes about TWO HOURS for anything you drink to have an effect on your body’s hydration level. By the time you feel thirsty, you are two hours behind! Drinking something even the second you realize you are thirsty means that you will still be dehydrated for at least two more hours. You should be drinking enough liquids throughout the day so that you rarely (or never) feel thirsty.
Here is an example of the other side.
Medically, dehydration is defined as a 5% increase in the concentration of solutes in your blood. (Often this can be more conveniently detected based on short-term weight loss.) Thirst sets in at about 2%, so you'll always feel strong thirst setting in long before you're dehydrated.