As James Smith points out, Dogberry is significant to the plot because he actually represents the play's most central themes, that of excessive pride or arrogance and appearances vs. reality. The fact that it is Dogberry and the watch who uncover Don John's plot against Claudio and Hero is significant because, it not only points out Dogberry's inconsistencies, it points out all the characters' inconsistencies as well.
Dogberry's vain stupidity actually mirrors the vain stupidity found in the characters that are the rulers of Messina. Both Prince Don Pedro and Governor Leonato have their absurdities. Don Pedro shows up in the first scene with his illegitimate brother who has just tried to overthrow his crown, saying that he and his brother are now reconciled, as we see in Leonato's lines, "Let me bid you welcome, my lord. Being reconciled to the prince your brother, I owe you all duty" (I.i.132-133). As it has been pointed out, when Elizabethans heard of an attempt to overthrow the crown, they would have sounded the alarms ("The Artificiality of Messinian Society," eNotes). Elizabethans would have known from their own history that the only way to deal with Don John and prevent future uprisings would be to have him executed. Don John, a person of questionable character, should not have been "reconciled" to his brother. Since Don Pedro has allowed someone who has tried to usurp him remain under his care, it brings his own character into question. Why would Don Pedro have decided not to severely punish his brother? Could it be that he wants to keep up appearances, just like we see other characters of the play do? If so, then it raises questions about his pride and vanity and shows us that, like Dogberry, he has an excessive amount of both and he is not the leader he seems to be.
Similarly, we can also question the virtue of Leonato's character as a governor, especially because it is he who put Dogberry in his position as Constable. We especially see his inefficiencies as governor when Dogberry and Verges come to report that they have captured criminals. Had Leonato put the protection of the city first and agreed to witness the criminals' interrogation that morning before the wedding, he would have learned about Don John's plot sooner and would have been able to spare Hero from such torturous humiliation. However, caring more about the appearances of the perfect wedding, he decided to set vital matters of the city aside, showing us just how much Dogberry resembles even the governor of Messina in vain stupidity.
Dogberry and the watch make it quite clear that, while they take great, absurd pride in their jobs, they actually have no intention of really doing their jobs. Instead, they intend to let "vagrom men," or "vagrant" men go, would "rather sleep than talk," and let thieves "steal out of [their] company," showing us just how poor they are at their jobs and that though, through his pride, Dogberry would appear to be good at his job, the reality is that he isn't (III.iii.22, 34, 55). Hence, it is the height of irony that it is the watch who catches Borachio and Conrade when that is the exact opposite of what they intended to do. Since it was Dogberry rather than any of the other leaders who discover the treachery, this inconsistency shows us just how inconsistent the leaders of Messina are, portraying the themes of both pride and appearances.
Dogberry and the Watch are primarily there to be comic releif. However, they serve as the vehicle which provides Don Pedro and Claudio with the knowledge of the plot to slander Hero. This is a pivitol point in the play that allows for Don Pedro and Claudio to right the wrongs they have caused and to seek forgiveness for their deeds. All of this allows for the play to end in the traditional manner for a comedy: a wedding.