There's probably no definitive answer as to why Esperanto uses a verb for that, just as there's probably none to why English uses an adjective for it. (For why both are possible in Esperanto, see Neil's anser.)
Languages differ in how they express certain concepts. They even differ in what concepts they ascribe to situations. Is "being right" a property (trait) of the person? (That's what adjectives are usually used for.) Or is it related to what they do (e.g. say or write)? (That's what verbs are usually used for, sometimes in combination with adverbs.) It's difficult to give a neutral answer to these question that isn't primed by one's native language or other languages one's familiar with.
While English deemed1 an adjective more fitting for describing that someone states or stated something that's correct, Esperanto favors1 a verb, but also allows for the adjective, if one prefers. (And as Neil noted, the adjective is actually the base form of the word / root prav-.)
And those aren't even the only options: German expresses that notion with "Recht haben" so seemingly applies the concept of possession instead of that of action or trait, and uses a noun together with "haben" ("to have (got)"). French's "avoir raison" also works that way, but French also seems to have "dire vrai" to express more or less the same.
Just like any living language, Esperanto isn't just a copy of another language (e.g. English) with only words substituted. It has not only its own grammar, but also its own idioms and expressions. While Esperanto was designed to be more logical than many ethnic languages, so that it'd be easier to acquire and so one can often deduce the correct phrasing from the wanted semantic and vice versa, there are statements that cannot be made without choosing some concepts to express them, and the choice made in Espranto can seem (and not seldom is) arbitrary when seen from the perspective of other languages.
1 Off course languages don't themselves make decisions. It's the users of the language past and present, and in the case of planned languages, the creators, who collectively consciously or unconsciously made and make those decisions. I'm phrasing this metaphorically / allegorically.