Don Pedro and Claudio believe the accusations because the of the "proof" that has been presented. Believing that Margaret is Hero based on the view the "see", it is easy to accept Don John's word. This is a large theme in this story and many of Shakespeare's stories, the problems associated with communication and with assumptions. It was, in Renaissance time, referred to as "noting". Here is a summary:

Noting
In Shakespeare’s time, the “Nothing” of the title would have been pronounced “Noting.” Thus, the play’s title could read: “Much Ado About Noting.” Indeed, many of the players participate in the actions of observing, listening, and writing, or noting. In order for a plot hinged on instances of deceit to work, the characters must note one another constantly. When the women manipulate Beatrice into believing that Benedick adores her, they conceal themselves in the orchard so that Beatrice can better note their conversation. Since they know that Beatrice loves to eavesdrop, they are sure that their plot will succeed: “look where Beatrice like a lapwing runs / Close by the ground to hear our conference,” notes Hero (III.i.24–25). Each line the women speak is a carefully placed note for Beatrice to take up and ponder; the same is true of the scheme to convince Benedick of Beatrice’s passion.
Don John’s plot to undo Claudio also hinges on noting: in order for Claudio to believe that Hero is unchaste and unfaithful, he must be brought to her window to witness, or note, Margaret (whom he takes to be Hero) bidding farewell to Borachio in the semidarkness. Dogberry, Verges, and the rest of the comical night watch discover and arrest Don John because, although ill-equipped to express themselves linguistically, they overhear talk of the Margaret--Borachio staging. Despite their verbal deficiencies, they manage to capture Don John and bring him to Leonato, after having had the sexton (a church official) “note” the occurrences of the evening in writing. In the end, noting, in the sense of writing, unites Beatrice and Benedick for good: Hero and Claudio reveal love sonnets written by Beatrice and Benedick, textual evidence that notes and proves their love for one another.

Don John tells us why he hates Claudio in Act 1, Scene 2:

That young start-up hath all the
glory of my overthrow: if I can cross him any way, I
bless myself every way.

Don John is something of a malcontent, and hates almost everyone; however, he has some special hatred in his heart for a few people.

The first is his half brother, Don Pedro, the prince and heir to the throne that Don John has just tried unsuccessfully to steal. Claudio is his half-brother's right-hand man. He is young and, according to Don John, arrogant and undeserving of power (he calls him an "upstart").

What has really angered Don John is that he thinks that Claudio has been responsible for and has benefited from his overthrow. So anything he can do to make Claudio's life miserable will be pleasing to Don John.