Winthrop, as one of the early governors of Massachusetts, was referring to the establishment of the city of Boston, which in early colonial times was geologically comprised of 3 hills.  The tallest of these, although now one third shorter, remains known as Beacon Hill, where the earliest colonists settled.  The top of this hill actually served as a beacon; a bucket of pitch was ignited and hoisted upon a pole as an alarm signal to the Puritan Community, similarly as the continual ringing of church bells became a century later. Being a theocracy, Winthrop was alluding to the Sermon on the Mount passages in the New Testament as previously mentioned; however, he was also alluding to the Holy City of God's chosen, or the New Jerusalem, described in Revelation, where the city was illuminated only by the presence of the Almighty for the rest of the world to see.

For we must consider that we shall be as a city upon a hill, the eyes of all people are upon us.  So that if we shall deal falsely with our God in this work we have undertaken, and so cause Him to withdraw His present help from us, we shall be made a story and a byword through the world...

These words of John Winthrop are quoted by President Ronald Regan on January 25, 1974, at the first Conservative Political Action Conference shortly after the return of John McCain, Bill Lawrence, and Ed Martin, POWs from North Vietnam.

After praising America's achievements, Reagan castigates the increase in government's size, power, and cost.  He praises the conservative group, saying that their thinking is more in accord with the hope and aspirations of "our people than are those who would sacrifice freedom for some form of security." Reagan, then, cites the lines of Winthrop near the end of his speech, and declares,

Well, we have not dealt falsely with our God,....We cannot escape our destiny nor should we try to.  The leadership of the free world was thrust upon us two centuries ago in that little hall of Philadelphia. In the days following World War II, when the economic strength and power of America was all that stood between the world and the dark ages, Pope Pius XII said, 'The American people have a great genius for splendid and unselfish actions.  Into the hands of America, God has placed the destinies of an afflicted mankind.'

We are, indeed, and we are today, the last best hope of man on earth.

Reagan's last lines explicate the meaning of Winthrop's words in modern times:  America must help to maintain freedom in this world.

Another interpretation of Winthrop's saying links to the belief of American Exceptionalism. In this line of thought, America and its history is placed in an "exceptional" light, suggesting that its narrative sets it apart from the rest of the world.  People who point out the notion of "a city upon a hill" might suggest that America's legacies to the world make it distinctive.  Its history of constitutionality, the longest running democracy, the presence of individual rights as part of its foundation, and its firm endorsement of capitalism and free market enterprise make it "exceptional" as opposed to other nations.  It is with this exceptional nature in mind that America and its values can be seen as a "city upon a hill."

When you say something is a "city on a hill" it means that it is out there in plain view for everyone to see.  The reason that that matters is because that thing in plain view will be an example to everyone else about how they should live.  So when politicians say that the US is a city on a hill they are saying that we are an example of how other people should be.

As you say, that goes back to the days of the Puritans.  They thought their society should be a city on a hill to show everyone else how to live in the way that God wanted.  We don't always mean it in a religious sense anymore.  Now it can just be an example of how to have a good government or something like that.