I think John is correct; to elaborate on what Bernie says on this:
"So the next time you hear me attacked as a socialist, remember this: I don't believe government should own the means of production, but I do believe that the middle class and the working families who produce the wealth of America deserve a fair deal."
The contents of this speech were very similar to other statements he has made about socialism across his entire political career. The entire speech could have been summed up neatly in a quote he gave to the Associated Press back in 1997:
"To me, socialism doesn't mean state ownership of everything, by any means, it means creating a nation, and a world, in which all human beings have a decent standard of living."" But that quote is alas fairly dated.
That would be social democracy to any European.
Social democracy is a theory of democracy that, in line with the social part of Human Rights, defines the welfare state as an essential component of any functional democratic state. It applies a wide range of economic and social interventions to promote social justice as both prerequisite and target of a well-functioning liberal democratic polity and a capitalist economy which serve the general interest. Social democracy is committed to representative or participatory democracy, measures for income redistribution, legal regulation of the economy, and welfare state provisions. Social democracy thus aims to create the conditions for capitalism to lead to greater democratic, egalitarian and solidaristic outcomes
And if the obvious fact needs more authoritative sources, here's The Economist's take on Bernie's (2016) platform:
BARACK OBAMA spent his first campaign for president explaining why he was no socialist. Bernie Sanders, the out-of-nowhere candidate who has constituted an unexpected challenge to Hillary Clinton, bravely embraces the label. Yet while Mr Sanders has built his campaign on a jeremiad against wealth inequality and corporate greed, he isn’t, properly speaking, a socialist—or even a democratic socialist. The better term encapsulating Mr Sanders’ positions is “social democrat”, a label that jibes with his rather mainstream embrace of “private companies that thrive and grow in America” and belief that “the middle class and the working families who produce the wealth of America deserve a fair deal”. To clarify matters, Mr Sanders flatly disavows the very heart of socialism as defined by Karl Marx: “I don’t believe government should own the means of production”, he says.
As for why he is receiving support of actual socialists like DSA, etc.
In the socialist milieu, the proponents of building a campaign for “Bernie 2020” put forward two main arguments: first, that Sanders’ campaign is the “best available means to raise workers’ class consciousness,” and second, that there is a battle inside the Democratic Party between a progressive and a corporate wing, and that we socialists need to take part in that fight.
And some US socialists don't actually agree with that strategy.
Sanders’ politics have often been described as “populist left,” and with good reason. His social-democratic policies are blended with nationalist rhetoric about “American values” and “keeping good jobs.” Peter Frase, writing for In These Times, makes the point that rallying behind Sanders’ popularity can “obscure the need to ground our struggles in mass organizing” and lead us to wed ourselves to his “New Deal liberalism” rather than debating what “socialism” really means.
Since the term "democratic socialism" has little traction outside of the US, it helps to just explain the DSA's initial vision:
When the DSA was founded in 1982 by Michael Harrington [...] Its organizers believed it could be more useful as what some have called basically a lobbying group to the Democratic Party, pushing the mainstream liberal party further left. Harrington described his group’s stance as “the left wing of the possible” and he knew from first-hand experience that such influence could be powerful: the Great Society programs of Lyndon Johnson’s presidency were partly inspired by Harrington’s book The Other America: Poverty in the United States.
As for DSA's own platform, it is further to the left than Sanders', e.g.
As we are unlikely to see an immediate end to capitalism tomorrow, DSA fights for reforms today that will weaken the power of corporations and increase the power of working people.
Sanders is not a member of DSA, by the way. So there is in this sense, a struggle over the meaning of "democratic socialism" in the US.
In a broad sense
The term democratic socialism is sometimes used synonymously with socialism, but the adjective democratic is sometimes used to distinguish democratic socialists from Marxist–Leninist-inspired socialism which to some is viewed as being non-democratic in practice.
But this leaves a wide area of ambiguity as to what it is actually about. Which is why the Wikipedia article on this topic is a kitchen-sink of every movement remotely socialist that endorses some form of democracy, ranging from the Sandinista National Liberation Front, Syriza, the Labour Party under Jeremy Corbyn, Mélenchon's La France Insoumise, Canada's NPD, etc.