One morning in August, when I was a guest on the C-Span television network, I got a phone call that took my breath away. "I'm a white male," said the caller, who identified himself as Garry from North Carolina. "And I'm prejudiced."

As a black leader often in the media, I have withstood my share of racist rants, so I braced myself. But what I heard was fear - of black people and the crime he sees on the news - not anger.

"What can I do to change?" he asked. "To be a better American?"

I thanked him for admitting his prejudice, and gave him some ideas - get to know black families, recognise the bias in news coverage of crime, join an interracial church, read black history.

In a professional capacity, I typically speak about race in terms of law and policy. But with this man on the phone, it felt right to speak to the basic human need I heard in his voice: to connect.

The video of us went viral, surpassing eight million views. After a racially charged summer, a lot of people saw something they hungered for in our exchange. To white viewers, here was a black woman who was morally clear but not angry. To people of colour, here was a white man admitting his racism - finally.

Garry found me on Twitter after our televised call, and, shortly before the presidential election, I visited him near his home town in the Appalachian foothills of North Carolina. We met on a patio amid the changing fall colours, as his dog kept a watchful eye nearby.

There we were: Two products of this country who couldn't be more different, having the oft-invoked but seldom practised "conversation about race". I was surprised when he said that he had followed my suggestions and was dedicated to "getting right about this before I die". He talked about the fear he carried towards people of colour, and how it had become a physical weight. "It's killing me on the inside," he said. "If I don't change things, I could have a stroke."

Although Garry didn't vote for Mr Donald Trump, he is the media's image of a Trump voter: a rural, middle-aged, white male from a working-class background. "We're a troubled group right now," he said to me when we met. "We're not a growing part of the population, we're diminishing. I think our culture is mixing real fast. Instead of the usual 20 years it takes to change society, it's happening in five years. It feels like an overwhelming wave is rushing over us."

Research shows that when white people become attuned to demographic change, they become more conservative. The right-wing narrative is that such change is the unmaking of the United States. I told Garry I believed it was the fulfilment of our country.

We talked about what it would mean to be the "better American" he invoked on our call. I said that person would be able to find common cause with people of all backgrounds. Garry's eyes brightened, but he said that it would take time. "I speak for a lot of unspoken people," he told me; "maybe millions of white people who are afraid to admit" their racial fears and prejudices. "They're not bad people. They just don't know how to behave and how to interact" with people of different races.

Garry had some advice for me, too. "Talk to white people," he said. "We need a little bit of guidance. We're not really getting it from our politicians. They want to play one side against the other for votes."

We steered clear of politics in our first few conversations but, after the election, he was eager to talk. Garry saw Mr Trump as the peddler of all the toxic ideas about people of colour that he now avoids on television. (He said "watching too much TV" is something he has in common with both Mr Trump and his voters; we did meet, after all, through a C-Span call.) He thinks many Trump voters could benefit from the journey he's taken.

With Mr Trump headed to the White House, my now-friendship with the "racist caller" on C-Span seems like a glimpse of a path not taken. Garry makes me believe that even though a man endorsed by the Ku Klux Klan won the majority of white support, people can change. He told me he now notices his own stereotypes and is eager to replace them with something more generous and true about his fellow Americans.

We need conversations like mine and Garry's to happen across the country, outside of politics. Societies that have been through traumas have embarked on racial reconciliation processes; South Africa's is the most famous, but there are dozens more.

There's no reason we can't do that here. Demos, the think-tank I run, is working with a variety of other groups on such an effort in 10 communities next year. I spent the past week meeting hundreds of people - librarians and teachers, community organisers and police officers - who are preparing for conversations in their communities.

"What can I do to change?" Garry asked when he called in this summer.

I was able to answer him because he had first acknowledged what so many people deny: the persistence of prejudice. That's the first step for all of us to become better Americans.

• The writer is president of Demos, a public policy organisation.