I am not an anti-realist, but since none stepped up I'll try to explain as I understand it. Dummett, the founder of modern analytic anti-realism, emphasizes that unlike realism, anti-realism is not a unified doctrine, one can be an anti-realist about some specific domain (mathematics, physics, ethics, past, future, etc.), and a realist about the rest. The motive behind it is that for statements to have meanings they must be understandable, and understanding them means being able to access evidence for or against them because meaning can only be communicated to others overtly.
Dummett's original position is expressed in Reality of the Past, and is simply that all we can mean by claims about the past reduces to organizing traces of it in the present, or rather of what can be usefully treated as "traces of the past". Under his moderate version of anti-realism, justificationism, we do not commit to "reality" of such claims, but neither do we deny it. For instance, consider "king James II had migraine on the day of his 32 birthday". If there is a record of it somewhere then we know how to understand it, but what if there is not? What does it mean exactly? That it "really" did happen out there? But if we can have no access to out there in principle it is questionable that we "really" understand what that means. How then can it be confirmed or refuted? It would seem that we are implicitly imagining "in principle" something like a time machine. What about "Hilbert had discalculia"? Discalculia is a learning disorder that was only identified in 1974, Hilbert passed away in 1943. Even more than a time machine would be needed to make sense of this one.
The more concrete a statement, the less it is removed from the present, the stronger the "feeling" that we "really" do understand it regardless of access. But as statements get more general and further removed their meaning reduces to little more than this is a useful way to organize available trace record. And on the available record there is little distance between realists and anti-realists, but anti-realists do not attach any metaphysical add-ons to it, and reserve judgement on the unknowable. So "the very first mammal appeared in Africa" has no logical value, true or false. In practice, back projection of trace statements is a handy mental shortcut, the same one we routinely take with everyday objects or even with theoretical entities like atoms, the same one our visual cortex takes to fill in patterns. But a shortcut is all it is, and it produces illusions just as well as it detects patterns.
Is this counterintuitive? Very. Dummett himself after 40 years of anti-realism called it "coherent but repugnant", and experimented with softening justificationism about the past by placing it into subjunctive mood. So the past tense is meaningful if "someone suitably placed could have verified it". Presumably, "suitably placed" allows for something like a time machine to do the placement, so "king James II had migraine on the day of his 32 birthday" would clear the hurdle, but I am not so sure about the aspects of Big Bang not recoverable from the current universe. See Rispberg's Objectivity of the Past.
There is a stronger version of anti-realism, called contructionism, which goes beyond justificationist reservation of judgment, and positively claims that history is socially constructed rather than "real", and even the terms in which it is expressed only mean in the cultural context of the constructors (and not what they ostensibly purport to mean). This is more often applied to human rather than natural history, and is a common theme in social and cultural criticism. Orwell immortalized a sinister version of it in 1984, "he who owns the present owns the past; he who owns the past owns the future", which was approximately practiced in some communist and fascist countries. See more in Pataut's Anti-realism about the Past and Burr's Introduction to Social Constructionism.