Roderigo was becoming aware of the fact that the huge sums of money he had given Iago to help seduce Desdemona had been going directly into Iago's retirement fund. When a conman cannot blow off the mark, either he runs or arranges for him to be killed. Actually Iago wanted Cassio to kill Roderigo but he didn't quite do the job so Iago has to finish it.
Iago advises Cassio to ask Desdemona to plead on his behalf. This will help Iago poison Othello's mind, as Desdemona will appear to be doing so out of her (illicit) love for Cassio.
He tells Cassio to ask Desdemona for her help to get him reinstated.
Iago says that he was sleeping next to Cassio and that Cassio was having a dream about making love to Desdemona.
She found a handkerchief, and iago wanted her to copy it.
Iago tells Cassio that he must persuade Othello to reinstate him. He says "the general's wife is now our general" and so the best way to persuade Othello is to get Desdemona to plead on Cassio's behalf. Since Iago has also told Othello that Desdemona and Cassio are having an affair, the harder Desdemona pleads, the worse it is going to look for both her and Cassio.
Iago realizes that Cassio greets Desdemona romantically, and with a kiss on the cheek. Supposably that's how you greet the women politely.
Othello was just out of earshot when Iago talks to Cassio. He has told Othello that he is talking to Cassio about Desdemona but he is really talking about the woman that Cassio is really having an affair with, Bianca. In 'Othello' Iago tricks Othello by planting Desdemona's handkerchief in Cassio's room.
Desdemona. Iago says that Desdemona is the most influential person with Othello, so Cassio should get her to plead on his behalf. This sounds reasonable, except that Iago has suggested that Cassio and Desdemona are having an affair, and this will heighten Othello's suspicions.
Cassio sees Desdemona as a sort of secular Virgin Mary, as from the quote, "O behold...Hail to thee lady!". On the other hand, Iago sees Desdemona as competition for Othello's affections.
If Desdemona asked a favour on Cassio's behalf, it would look to Othello like she was doing it out of an improper love for Cassio. The harder she argues, the more suspicious it seems to Othello.
Primarily the fact that Iago has suggested to him that that is what is being talked about. By this point, Iago has got Othello into such a state he cannot interpret anything Cassio or Desdemona does as being innocent.