The 60% figure is a reference to the votes necessary for cloture (i.e., to end a filibuster) in the Senate, but I don't think Obama meant the point in a strictly technical sense. After all, given the equal representation of each state in the Senate, you can win the support of more than 60% of the People and still not have 60% or even a majority of the seats in the Senate. I think that what Obama had in mind was something a little different.
Political scientists have noted how, in recent elections, the major parties have pursued a strategy of polarizing the electorate and then aiming to turn out the base, rather than what for years had been the standard strategy of trying to appeal to the median voter. The new strategy has, in turn, contributed to the appearance of a highly polarized electorate. Yet while there are regional differences of opinion on various policy issues, these tend to be small. (For a good account of how small, see Culture War? The Myth of a Polarized America, by Morris Fiorina, Samuel Abrams and Jeremy Pope.) Only by making these issues salient precisely along the axis of greatest disagreement do politicians pursuing a polarize-and-turn-out strategy create the appearance of a red state America and a blue state America. Obama offers the promise of a kind of politics in which the cultural divide (gun rack on red-state pickup versus save the whales sticker on blue-state volvo) is simply not relevant to the great issues of the day.
Whether anybody can accomplish the sort of political transformation/realignment that Obama seeks remains to be seen, but he pretty clearly is better suited for the job than either of the other top-tier Democratic candidates. Edwards, in talking about two Americas, is not trying to polarize, and certainly does not imagine that the divide between the two Americas is red/blue; it's rich/poor. But the very trope invites a kind of polarization. Meanwhile, Clinton's anti-base of about 40% unfavorables means that right off the bat she needs to hold the base to win election, and even though she has been a fairly centrist Senator and as much of a triangulator as the first President Clinton, she is, fairly or not, a polarizing figure. By contrast, Obama, from his first entry onto the national political stage, has been deriding the idea of a polarized America. In the wake of Howard Dean's controversial but successful midterm election strategy of trying to compete nationwide rather than securing the base and going after the most winnable swing states and districts, the Democratic party may be coming to realize the wisdom of a depolarizing strategy.
George Bush said he was "a uniter, not a divider," but Barack Obama means it. And that's the closest Dorf on Law will come to endorsing a Presidential candidate!