Why did the developers make Dexterity add to damage on DEX-based attacks? Are there statements by the writers/developers describing why DEX bonus is on damage done as well as to hit rolls? Have they said anything about how to justify your DEX applying as a damage bonus? Is it like a vital strike?
One reason is implied by an answer from Jeremy Crawford, lead rules designer for DnD 5e:
You always add the appropriate ability modifier to the damage of your weapon attacks, unless a feature tells you not to.
Thus, the implied intention of adding DEX modifiers to damage isn't unique to DEX. Rather, any weapon that is based on a particular ability gets its bonus as well. Thus a weapon that relies on STR gets bonuses to attacks and damage too.
Intuitively, for items like rapiers which use DEX, this fits the idea that not only can a character strike more accurately - but that skill translates into better damage. Presumably this reflects the idea that DEX based characters, like rogues and assassins, achieve their higher levels of damage through targeting vulnerable and critical areas rather than brute strength. Any weapons based on intelligence would also have higher accuracy and damage with highly intelligent characters unless otherwise documented as Jeremy notes.