Why must we add an 's? Why "she is at the dentist's now" instead of "at the dentist"?
We don't have to. It's equally correct to say
She is at the dentist now.
(you can interpret "the dentist" as a synecdoche in which the person stands in for the place) or
She is at the dentist's now.
(an elliptical way of saying "She is at the dentist's office now*).
Which you choose probably depends on what you hear more.
Adding the 's to make "dentist's" indicates the place owned by the dentist - in this case the dentist's surgery.
It is common to say that you are "at" a place - e.g. at the office or at the railway station. So "at the dentist's" is correct.
It is not common to say that you are "at" a person. "She is with the dentist now" would sound better.
While most people would recognize "at the dentist" as not quite correct, it is used quite often simply because it is easier (and neater) to say. "Dentist's" can get juicy, especially if you spend a lot of time at the dentist's getting bridgework on your front teeth. Americans are really conscious of spitting when they talk.(US)
Grammatically, "dentist" is being used as a placename here, referring to the office itself. That trailing "'s" was phased out of day to day usage in many places, many years ago. Specifying "dentist's" in some regions of the United States would sound, or be considered, overly fussy or exact. Probably not enough that someone would mention it to you, but enough that it would impart a tone to your conversations. This is a natural process of language becoming more efficient, as words take on contextual meanings.