1) Self-reported prejudices have dropped because the level of prejudice has diminished. Straight-forward enough.
2) Self-reported prejudices have dropped because White people have learned to lie about their prejudices. They know racism has become unacceptable so they don't admit to it, but actual underlying attitudes haven't changed that much.
3) Self-reported prejudices have dropped because conscious prejudice has diminished, but it has been replaced or overtaken by subconscious or implicit biases which now exist in tension with conscious (and sincerely-held) egalitarian beliefs.(I say this might be 2.5 explanations because #1 and #3 are consistent with each other).
The reason that the face-value "racist self-identification has fallen because racism is diminishing" argument can't be accepted uncritically is that there is ample evidence that some form of racially biased behavior is still rampant in our society. So alternatives two and three explain that behavior by either saying "White people are doing what they've always done and thinking what they've always thought, they've just gotten craftier in hiding the ball" (explanation #2) or "there's been an actual change in White attitudes, but the continued presence of racially biased patterns of behavior can be explained by the operation of subconscious biases" (explanation #3). We can characterize this split as between covert biases (known and internally acknowledged by the holder, but hidden) and implicit biases (unknown and internally rejected by the holder, but nonetheless still operating subconsciously).
One way of distinguishing between the covert versus implicit bias explanations is that they have different predictions about when prejudice will reveal itself. Covert bias suggests that people conceal their biases because it is socially unacceptable to reveal them; hence, such biases will emerge in social contexts where it is considered okay to express prejudiced views. Implicit bias, by contrast, suggests that people are trying to harmonize conscious egalitarian commitments with their subconscious prejudices; hence, bias operates in cases where the discriminatory behavior can plausibly be justified as being consistent with an egalitarian ethos (for example, one could favor a white job applicant against a black competitor in situations where they were roughly equally qualified by relying on "intangible" differences that are hard to measure; but one couldn't reject a black candidate with obviously superior qualifications because that would be self-evidently discriminatory).
Most of the work I've done in the discrimination context has essentially assumed we mostly live in the implicit bias box. I believe that in general people today honestly and sincerely buy into egalitarianism, but that this commitment coexists with subconscious prejudice. There's some evidence to support this hypothesis (Gaertner and Dovidio's "aversive racism" work being the most influential on me), and as a historical matter it can be derived from Gunnar Myrdal's description of the "American Dilemma". But there was always evidence that supported the covert bias hypothesis: for example, "bogus pipeline" studies (roughly, ones where researchers informed subjects that "we'll know if you're lying because we're scientists with fancy machines") have erased much of the drop in self-reported prejudices. That supports the covert bias hypothesis over the implicit bias one, because in general on the former the bias is known to the test subject (who then tells the truth because "the jig is up"), whereas someone with implicit biases may not even be aware he holds them and thus would be unaffected.
So which is it? Are there really significant portions of Americans who quietly harbor nakedly, unabashedly prejudiced views -- only keeping them quiet because they know that's what society demands?
Enter Donald Trump.
Donald Trump's continued appeal even after a string of outrageous comments -- ranging from Obama birtherism to Megyn Kelly "blood flowing from her wherever" to Mexican immigrants all being rapists to most recently nodding along to a supporter who wants to "get rid of" the Muslims -- has perplexed legal commentators who expect Republican primary voters to eventually recoil against such obvious forms of prejudice. That expectation is very much consistent with the implicit bias hypothesis: as it becomes increasingly difficult to square Trump's statements with egalitarianism, people should start to reject it.
But things are different from the covert bias position. When Donald Trump implies Obama is a Kenyan-born Muslim, when he suggests that Latinos are all violent criminals, when he legitimizes mass expulsion (or worse) of American Muslims -- maybe he's saying out loud what many people secretly believe but felt constrained in saying. Isn't this the root of the "anti-PC" backlash? "I used to be able to openly degrade women for having a menstrual cycle, but thanks to liberal elites and Feminazis I can't say that anymore! What happened to freedom in America?" The complaint of the anti-PC crowd is precisely that they have to keep quiet that which they'd rather broadcast (and once could broadcast, before we had to actually start listening to the desires of pesky minorities).
Most people can't say such things anymore, or at least they're constrained in their ability to do so. There are members of traditional outgroups in their workforce (maybe even their boss), or as powerful constituents, or major donors, or simply well-connected citizens. Saying such things comes with real costs, sometimes prohibitive costs. It can lose you your job, or your friends, or your reputation, or your candidacy. And some people resent that deeply even as they quietly stew and keep their true beliefs private.
But Donald Trump is different. He can say these things. He can't destroy his reputation -- he's his own brand. He can't be driven out of the race by outraged donors -- he doesn't need them. He can't lose his job -- he runs his own company. He doesn't have to defer to outraged outgroups -- what can they do to him? For someone with implicit biases, this may not matter -- he's so obviously over-the-top that his positions can't be reconciled with any sort of egalitarian commitment. But for the covertly-biased, he offers up a tantalizing vision where one can say all of those open, overt, explicitly biased things they genuinely believe and it's okay. They don't have to cover it up anymore. After all, someone with covert biases only keeps them covert because of social conventions which demand it. And social conventions can change -- most obviously, if the President of the United States, or even a mainstream presidential campaign with significant support and staying power, busts the taboo.
Bernard-Henri Levy once referred to the reemergence of anti-Semitism in terms of people who "feel once again the desire and, above all, the right to burn all the synagogues they want, to attack boys wearing yarmulkes, to harass large numbers of rabbis, to kill not just one but many Ilan Halimis...." This is what the Holocaust took away from the world, especially the European world: the ability to hate Jews and feel justified and righteous in doing so. And under this view, anti-Semitism never went away, it just was forced underground, and what people thirst for is for something to return what they see as their God-given (often literally) right to openly hate Jews without losing social status (many consider the flourishing of anti-Zionist ideology to be related exactly to this desire). And likewise we can say that the civil rights movement, the laying bare of Jim Crow and American apartheid, and the rise in the power and influence of former outgroups have all taken away the previously-enjoyed rights of straight White male Christian Americans to feel and express their superiority.
The incredulity over Donald Trump's continued appeal relies on the assumption that his supporters can't really buy into his extreme statements or his openly bigoted views. It follows, then, that if his following does continue to stay true, then maybe its not a case of people overlooking, or minimizing, or downplaying those views. Donald Trump is popular because of, not in spite of, his decision and ability to be avowedly prejudiced and not be drummed out of the mainstream.