The main reason that Cassius wants Brutus to join the conspiracy against Caesar is because Brutus is very widely respected among the people of Rome. Cassius does this by flattering and toying with Brutus' vanity.
The play "Julius Caesar" by William Shakespeare depicts the events preceding, during and after the murder of the Roman dictator Julius Caesar. In Act 1, Scene 2, Cassius first approaches Marcus Brutus, a Roman politician and very close friend of Caesar. Cassius begins flattering Brutus by asking whether Brutus can see how much the other politicians and the people respect him. When Brutus modestly answers that he does not, Cassius says he will take on the role of Brutus' mirror, so that he can see for himself.
When Cassius and Brutus hear people shouting for Caesar and express their mutual disdain that he might become king, Cassius sees his opportunity to seduce the well-respected Brutus to his conspiracy against Caesar. Therefore, Cassius begins questioning how unfair and unsafe it is that one person should tower over so many others. Brutus ends the conversation by saying that he is willing to consider Cassius' words. Brutus thus takes the first steps toward becoming a conspirator, thereby providing the conspiracy legitmacy because of its powerful, widely-respected figurehead.