Why and how carboxylate ion has greater stability than carboxylic acid?
First of all, it would have been clearer if it's a misprint or not if you had attached a screenshot of the wording in your textbook (you still can). 

At any rate, the subject of stability of carboxylic acids and carboxylate anions usually arises when discussing the level of acidity or in other words, the strength of the acid. It is pointed out in most organic chemistry books (where electron delocalization is explained) that electron delocalization or resonance is a concept that contributes to a certain structure's stability. That is to say, when a negative charge can be delocalized over several atoms (as is the case with a carboxylate group), instead of being localized on a single atom (as is the case with an alkoxide group), it is considered more stable energetically. (see here, section B - Resonance).

The relative stability of the carboxylate anion means that a carboxylic acid will favor (again, due to energetic considerations) the release of a proton in a basic environment (this is important, since in an acidic environment, the equilibrium will shift towards the acid, and much less carboxylate will exist in the solution), since the conjugate base (the carboxylate anion) is relatively stable, and therefore can exist in the solution in this form.

 (Taken from Organic Chemistry, by Paula Yurkanis Bruice, any edition really...)

To sum up, Carboxylic acids and their conjugate carboxylate anions are not (and cannot be) compared in terms of stability. Under the right conditions, i.e. in a basic solution, a carboxylate anion will be favored over a carboxylic acid because: 

a. There is a base to take a proton (be it a hydroxide ion, an aminic group or any other H+ acceptor).

b. The carboxylic group will release the proton rather easily, because what's left (the carboxylate anion) is relatively stable due to electron delocalization.