Puritans believed that the purpose of government was to assist the people in carrying out God's laws. Governments were the instruments by which a people executed their end of a covenant with God, and if governments interfered with the execution of God's laws, or acted in an ungodly way, it was the responsibility of godly men to reform or even overthrow the government. In short, Puritans had no concept of separation of church and state.
Puritans also believed that God would visit his wrath on a people whose government failed to act according to His will. This would later become the impetus for enforcing religious conformity in New England. But in England under Stuart rule, Morgan argues that Winthrop saw the signs of God's wrath everywhere:
This was...an England full of plenty and delights but under the shadow of God's wrath...In the 1620s, the textile industry suffered a depression that affected the whole country. Clothworkers were unemployed, hungry but unable to pay for country produce; clothiers could not market their fabrics; farmers could not pay their rents. The cost of caring for the poor and unemployed rose steadily. Was this not a hint of God's displeasure, a warning of worse to come?
The slump in textiles was especially frightening to Puritans, who dominated the industry. When Charles I took the throne from his father, things went from bad to worse from the point of view of Puritans like Winthrop. Charles was married to a Catholic, and sought to move the Church of England back to a more traditional approach that emphasized ceremony and the centralized authority of the church as a support for his own power. Charles's assault on Parliamentary power was viewed by Winthrop as a sign of the imminent decline of Protestantism itself. This they interpreted as a sign of God's wrath, and the only way Winthrop saw to avoid it was to leave England. Winthrop joined, and became the leader of the Massachusetts Bay Company, which departed for New England in 1630.
Source: Edmund S. Morgan, The Puritan Dilemma: The Story of John Winthrop (Boston: Little, Brown and Company, 1958) 18-33.