The plot against Malvolio is found in Twelfth Night (rather than Othello). In that play, Sir Toby Belch, Fabian, and Sir Andrew Aguecheek all take part in the revenge plot on Malvolio.
As far as why, it seems a combination of factors. The strongest is that Sir Toby finds Malvolio's manner (which springs from his character) irritating. They also do this because they can, and because they find it amusing.