On a snowy Saturday in Flatbush, the snowy-haired independent senator from Vermont kicked off his second bid for president. Around 13,000 supporters of Bernie Sanders packed into the Brooklyn College quad in Brooklyn, New York, on March 2. Some arrived as early as 8 a.m, though the candidate did not take the stage until 12:40 p.m. One canvassing table on Hillel Place, which leads from the subway station to the campus, displayed a banner that read “Trump Out, Bernie In.”

Tanya Covington, 48, cleans New York City subway trains. As several people got off at the Flatbush Avenue-Brooklyn College station, she shouted from the platform, “Bernie! Bernie! Bernie!” Wearing a bright orange vest, she fist-bumped attendees as they streamed past. Asked if she supports policies like Medicare for All, Ms. Covington said yes. “I support him trying to give a standard of living for everyone,” she said. “Not just the upper class.”

In the first speech of his campaign, Mr. Sanders reiterated the policy positions that defined his first run in 2016, among them support for unions, tuition-free higher education and single-payer health care. A son of Brooklyn, Mr. Sanders strolled onto the quad to the blare of Jay-Z’s “Brooklyn Go Hard.”

He and other speakers drew comparisons between Mr. Sanders and another native New Yorker, Queens-born President Donald J. Trump. “Brother Sanders…did not grow up with a silver spoon in his mouth like our president,” said Representative Terry Alexander of the South Carolina House of Representatives. Mr. Alexander also endorsed Mr. Sanders in 2016.

He and other speakers drew comparisons between Mr. Sanders and another native New Yorker, Queens-born President Donald J. Trump.

Mr. Sanders expanded on that point: “I did not come from a family that gave me a $200,000 allowance every year beginning at the age of three. As I recall, my allowance was 25 cents a week. But I had something more valuable: I had the role model of a father who had unbelievable courage in journeying across an ocean, with no money in his pocket, to start a new and better life.” Mr. Sanders’s father emigrated from Poland in the 1920s.

“We will no longer stand idly by and allow three families in this country to own more wealth than the bottom half of the American people,” Mr. Sanders said. “While these families become richer, over 20 percent of our children live in poverty, veterans sleep out on the streets and senior citizens cannot afford their prescription drugs.”

Mr. Sanders covered poverty, mass incarceration, climate change, military spending and education, among other topics. On abortion, he said: “When we are in the White House, we are going to protect a woman’s right to control her own body. That is her decision, not the government’s.” (Mr. Sanders drew some criticism from fellow Democrats in 2017 for campaigning for a pro-life Democrat in Nebraska.)

In introductory speeches, the Sanders team foregrounded race. Campaign surrogates Nina Turner (president of Our Revolution, a Sanders-affiliated political action organization) and Shaun King (a social-justice activist with a considerable social-media presence) discussed Mr. Sanders’s history as a participant in the civil rights movement of the 1960s, a critic of South African apartheid and a supporter of the Reverend Jesse Jackson’s presidential campaigns in 1984 and 1988.

“Brooklyn, you should be proud that the son of this city has been standing on the front lines for a very long time,” said Ms. Turner. “Standing up for working people in this country: black, white, brown, red, yellow and the swirl in between.”

Many Democrats are debating the idea of reparations for the descendants of slaves, but last week on the talk show “The View,” Mr. Sanders seemed to throw cold water on the idea: “I think that our job right now is to address the crises facing American people in our communities. And I think there are better ways to do that than just writing out a check.” Multiple candidates in the 2020 field—including Elizabeth Warren, Julian Castro, Kamala Harris, Cory Booker and Marianne Williamson—have endorsed some form of reparations.

Many Democrats are debating the idea of reparations for the descendants of slaves, but last week Mr. Sanders seemed to throw cold water on the idea.

Since the beginning of the year, about a dozen elected officials have announced bids or exploratory committees for the 2020 Democratic nomination, so Mr. Sanders joins an already crowded field. At the Brooklyn rally, Mr. Sanders’s surrogates sought to distinguish him from the pack, speaking of him as an unflinching activist-politician. Ms. Turner discussed Mr. Sanders’s role in the 2017 unionization of a Nissan plant in Mississippi and his recent criticism of Amazon: “When you are willing to look Jeff Bezos in the eye to say to the wealthiest man in the world that it is a sin and a shame, that it is rotten to the core, that an $11 billion company refuses to pay their workers a living wage and to get that $15 an hour—baby, that is the measure of a man.”

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Later, Mr. Sanders called for the development of a “humane border policy” and the extension of permanent legal status to DACA recipients, castigating the Trump administration for separating families at the U.S.-Mexico border. He also deplored the U.S. incarceration rate—the highest in the world—calling it an “international embarrassment” and saying public money should be invested in jobs and education rather than imprisonment.

Soundtracked by Muse’s “Uprising,” Willie Nelson’s “On the Road Again” and the Trammps’ “Disco Inferno” (with a particularly apt chorus: “Burn, baby, burn”) the rally attracted thousands of New Yorkers despite the bitter cold. Gobs of snow dropped from barren trees. The crowd wore jewel-tone beanies, corduroy pants and leg warmers. Someone clutched a sign with a white silhouette of Mr. Sanders’s head and the tagline “Abolish Billionaires.” Someone else built a snowman, then stuck a blue “Bernie” sign in its globular torso.

London Jamison, 27, sold cobalt and navy campaign T-shirts at a table, and when potential buyers commented that her shirts promoted Mr. Sanders as the “2016” Democratic Party nominee, she retorted, “That’s what should have happened. That’s what should have happened.”

Babbie Jacobs, 60, stood near the back of the crowd. “I believe in his policies and his message,” she said. “All the things he put forward earlier are getting traction….Some of these things may actually come to fruition, thanks to him.”

But David Curry, 28, and Tim Hone, 27, were undecided. Mr. Hone said that a primary season with many candidates will prove helpful to the Democratic Party. “It would be good to have a very robust primary where they fight it out to see who the best candidate is.”

Brandon Sanchez is an audience voices reporter at The Wall Street Journal. Previously, he was an O’Hare fellow at America.