It was a dramatic historical moment. A community of devoutly religious Christians traveled across the ocean to a relatively unknown land, radically different from the society they left behind. For these travelers, who called themselves Pilgrims, the journey took on spiritual as well as physical significance. When they arrived on the new continent, it was not simply the vast forests or lack of densely populated areas and developed commercial markets that made it a land of opportunity and rebirth. The New World represented a fresh slate, an ocean away from the sins and corruption of the Old World and a chance to start anew, to build a society from the ground up on firmly pious principles.
Native Americans' contact with these newcomers was a startling encounter across a vast cultural and spiritual divide. Not only did European settlement encroach upon their lands, but it brought devastating plagues of diseases never before encountered on the North American continent, which soon decimated Indian populations.
Both the English and the Indians survived the setbacks and the ravages of disease and death; different members of both groups approached one another with a range of reactions, from trust and generosity to hostility and suspicion. In their first half century on the new continent, the English settlers learned from, interacted with, and battled against the Indian nations of New England. They also developed the institutions for which they would be forever remembered: the town meeting, the Congregational church, the hard-scrabble farming life of New Englanders, and the Protestant work ethic, which influenced the character and composition of subsequent American societies.
If you are from New England, this answer is probably obvious. If you are from the South, the history of New England can show you how those late-comers (compared to the Chesapeake settlers) differed from the people who settled your part of the country. If you're from the Midwest or the West, you should know this stuff because frankly, almost everyone on the east side of the Mississippi River assumes that you don't. And, you might be in a unique position to look out from your geographic position (distanced both by time and place from Plymouth Rock) and think about the bigger picture: that is, how these earliest settlements have influenced modern American society, culture, politics, etc.
This story can also debunk a few myths that are still floating around, including: there were no slaves in New England (few of them, yes, but slavery existed there and was permitted by law); that this region was entirely consumed by religion (people made some good money in the port towns, and just wait until you get a load of the section on sex crimes); that the Puritans were stuffy, prudish, boring old fuddy-duddies (sure, they seem conservative by modern standards, but these people wouldn't have passed laws against intoxication if there wasn't some cause for it).
Besides, we should respect these men and women for their courage and their commitment; they traveled for three months on stormy seas with undoubtedly awful food and plenty of seasickness (and other kinds of sickness). They fled persecution in their native England to another continent so far away and so foreign to Europeans that it was commonly referred to as "the New World." When they got here, half of them died of disease, cold, or starvation. Instead of freaking out and sailing back with the ship, the rest of them stayed on. All for the sake of starting anew and adhering to the tenets of their cherished faith. Even if you aren't religious, that's pretty impressive.
If you come from a Native American background, if that is part of your ancestry, or if you are smart enough to be curious about the people who were here when the Europeans arrived, then this is the chapter of history for you. The "United States" was still a long way off when the Pilgrims arrived at Plymouth in 1620, but the origins of American society are here. It would be misleading to draw any direct lines from 1620 to 1776, but the point is, this country was not simply a transplanted version of Europe. There were important reasons why the New England settlers came across the Atlantic ocean; they had substantial differences with the church and government in their homeland. And after their arrival, this society was inevitably formed not purely from their acts and beliefs alone, but as a result of their interactions (both violent and peaceful) with the indigenous people of the region as well as the dissenters and other groups that they would come across over time. Colonial America was, from its beginning, a complex mixture of widely different societies, and its history is the essence of their intertwining stories.