How why did the Trojan war start?

Paris, the son of the King of Troy, seduced the wife (Helen) of a Greek king (Menelaus). Whether she went voluntary or not (Paris was supposedly the handsomest man ever) isn't well established, except that she looted the palace and took it off as a gift to Paris.

Menelaus was brother to Agamemnon, the King of Kings of Greece. Agamemnon then raised an army and attacked Troy, to bring back Helen and of course the loot. Of course the opportunity to destroy a rival kingdom and sacking the rich city of Troy played a minor :) role.