If you listen to the full interview – which runs 42 minutes, much longer than that 1 minute or so of soundbite – or just read the transcript, you'll quickly see that this wasn't Atwater suddenly baring his soul and confessing that Reagan was engaged in an extension of the "southern strategy." No, this is Atwater mocking a liberal academic and making his arguments sound like a weird conspiracy theory.
Before the passage that everyone knows, there's this section, in which Atwater readily admits that Barry Goldwater and Richard Nixon had a racist "southern strategy" – Harry Dent was a Strom Thurmond aide who became President Nixon's point man on the South – but then insists that the Reagan team in 1980 did nothing of the sort.
…so what you had was two things happening that totally washed away the Southern Strategy, the Harry Dent-type southern strategy. And that is…That whole strategy was based - although it was more sophisticated than a Bilbo or a George Wallace - it was nevertheless based on coded racism. The whole thing. Busing. We want a supreme court judge that will not bus. Anything you'd look at could be traced back to the race issue and the old Southern strategy. And it was not done in a blatantly discriminatory way. But the Reagans did not have to do a Southern strategy for two reasons: Number one, race was not a dominant issue. And number two, the mainstream issues in this campaign had been “Southern issues” since way back in the 60s. So Reagan goes out and campaigns on the economics and on national defense, the whole campaign was devoid of any kind of racism, any kind of reference.
I'll pause here to acknowledge what you're currently muttering to yourself:
Yes, the claim that Reagan – who launched his national campaign with a speech praising "states' rights" in Neshoba County, Mississippi, and more – was not actually engaged in a race-baiting "southern strategy" is utter nonsense.
But my point here is that in this particular conversation, Atwater is clearly not admitting that the Reagan campaign did all that. He's repeatedly denying it.
With that in mind, look at the full context around the initial passage above.
He starts off by saying, first, there's no need to pull that coded racism anymore because southern white conservatives were already pulled into the Republican camp when Nixon did it and now Reagan can just appeal to them on other issues.