The Puritans believed in what's called election, whereby God chooses some people to carry out a particular purpose. An obvious example from the Bible would be God's election of the Israelites as his chosen people.
Closely related to the notion of election is the Calvinist doctrine of double predestination, whereby some people are destined to be saved, while others are to be damned. According to Calvinists, this decision had been made by God before he even created anyone. As such, there was absolutely nothing that anyone could do to guarantee salvation; it was all in God's hands.
Winthrop and the other Puritans who landed in America believed themselves to be part of the elect. This gave them the firm belief that they had been instructed by God to travel across the Atlantic Ocean, with all the hardships that that entailed, to establish a godly commonwealth in the New World.
In his famous sermon "A Model of Christian Charity," Winthrop says that God loves the elect because they are like himself. When he looks upon the elect, he beholds his beloved son, Jesus Christ. Being part of the elect may be a privilege, but as Winthrop understood all too well, with privileges come huge responsibilities. The Puritans must at all times remain faithful to their sacred covenant with God, and that's easier said than done.