One way to look at the question of Hamlet's madness is to rephrase your question to ask why it was that Shakespeare caused Hamlet to act mad. This is a crucial difference, because we are not actually talking about the motivations of a real person, but about a character in a play, whose actions may actually be motivated for reasons of dramatic effectiveness.

Mad scenes were a standard part of the genre of the revenge tragedy in this period and provided opportunities for actors to display a range of dramatic abilities. Audiences also enjoyed them. On a plot level, Hamlet's feigned madness leads, with a sort of tragic irony, to Ophelia's real madness.

Finally, by pretending to be mad, Hamlet appears harmless – a lovesick fool rather than a methodical plotter or assassin – although, as Polonius figures out, there is a "method to his madness."