Under the social contract theory how do leaders gain the right to rule

By the consent of the people over whom they rule.

The "social contract" refers to an implicit agreement between a government and the citizens of the society overseen by that government.   Philosophers of the Enlightenment era were famous for arguing the idea of a "social contract."  According to this view, a government's power to govern comes from the consent of the people themselves -- those who are to be governed.  This was a change from the previous ideas of "divine right monarchy" -- that a king ruled because God appointed him to be the ruler.  One of the most influential of the social contract theorists was John Locke, who repudiated the views of divine right monarchy in his First Treatise on Civil Government.  In his Second Treatise on Civil Government,  Locke then argued for the rights of the people to create their own governments according to their own desires and for the sake of protecting and enhancing their own life, liberty, and property.