They saw someone they knew who told them as much as he knew, told them what he didn't yet know and what the city was doing about it.
When I saw Obama on TV amid devastation in New Jersey on Thursday, he was making a speech. Impromptu maybe, but a speech.
That has been the story of his presidency. Oratory carried him to the White House but hasn't sustained him in power. It has merely left the question for this campaign: is Obama more than a speech?
The first television debate was a disaster for him. To do better in the subsequent contests he had to go on the offensive, surrendering the noble heights of his first campaign to get down and do the old politics.
The young voters who heard a new, inspiring brand of leadership from him four years ago might not bother to vote for anyone this time. If they don't turn out for Obama in similar numbers this time, Mitt Romney will be President.
Obama is banking on that prospect being enough to bring them out. Romney has done his utmost to appear harmless to them and convince them he can give business the confidence to create jobs. Unemployment among Americans aged 18 to 29 remains much higher than the rate overall.
It is good to watch a United States presidential election from this distance, far from the fierce allegiances that election campaigns everywhere are designed to revive.
If I had a vote, I'd be torn. Romney might be able to jolt the world economy out of its doldrums with a dose of financial realism that would be painful initially but would let investors see a credible future.
Obama, though, has a quality that could be equally valuable to the improvement of the world. He knows foreign countries as few Americans do. Africa was his father's home and he knows it well. Few African-Americans can say that.
But it is through his white mother that he has gained his intimate knowledge of other places. Her field work took him to Indonesia for several years when he was very young. His playmates were from families that practised rites of Islam, Buddhism and Hinduism in the religious mix of that country.
Then he grew up in Hawaii. He knows Polynesia and the Pacific.
At universities in Los Angeles and New York he gravitated to foreign students and among his closest friends were two Pakistanis.
He credits them with explaining Muslim politics to him, especially the tensions between Sunni and Shia.
He went back to Karachi with them for a few weeks and saw the wealth and poverty of the country at close quarters.
His mother also lived in Pakistan, India and Bangladesh for periods of research and he visited her in each place. As a candidate and President he has not said much about these experiences, particularly the Muslim connections.
He was taking enough flak from Fox News and right-wing radio about his birth, nationality and middle name. But a second term would free him from that nonsense and, just possibly, his presidency could fly.
Like Romney on the economy, Obama has said little on foreign policy to support that hope. Their television debate on foreign issues was a typically blinkered American contest to back Israel and blockade Iran.
Oddly, Romney seemed to blame Obama for the success of a mildly Islamist party in Egypt's presidential election. Romney has no sense of the world outside America.
The 21st century will be shaped by the way civilised countries respond to a movement that would shoot a girl seeking an education.
War won't defeat it, sensitive influence might. On that front, Obama would be the better bet.
