In Platonism it is claimed that there exists abstract objects. But how can we come to know anything about them if they are not spatiotemporal and exist seperately from us?
Capital P Platonism holds that the physical world is derivative from the real world of forms. The mechanism for this process of spawning the physical is not specified. If Plato were publishing today he would likely treat the interaction as a subject for additional research, with the mechanism TBD and to be estabished thru investigation. Platonism is a proposed explanation for the reactions between ideas and the physical — it is a Popperian hypothesis.
Lower p platonism holds by the less ambitious claim that abstract objects exist. Most lower p platonists assume that abstractions cannot be causal. This assumption of lower p platonists spawns a highly relevant question of “so why would we have any reason to postulate their existence?” Under Popperian hypothesis forming there would be no reason, nor any way to test that assumption. Your question is more relevant for lower p platonism.
If one applies Popperian testing to the “no causation” assumption there are significant cases to question it. The “unreasonable effectiveness of mathematics in science” is consistent with math being causal in the physical world. This primacy of math and generative role is implicit in theoretical physics— where math processes explaining physics phenomenon is SOP. This generative role is asserted explicitly by several physicists, most notably Max Tegmark. It is also intrinsic in many of the interpretations of quantum mechanics, including both the informational and hologrammatic explanations.
One can also test for more mundane causation, where lots of ideas, such as the British Constitution, or love, are significantly causal.
If one accepts this causal interaction, then as with large P Platonism, HOW that interaction happens is TBD and subject to further study. This is a common status for many questions for pragmatists and empiricists. In pragmatic empiricism a wholistic answer about everything is still not yet available to us, hence our worldview must consist of only bits and pieces of well supported thinking, with many subjects still open areas of investigation that still need to be filled out somehow in our future.
As a pragmatic empiricist I consider many major questions to be in this TBD category. In addition to the nature of abstract objects and how they interact, I would list the nature of causation, the nature of time, and the nature of consciousness and how it interacts, the nature of emergence and how it happens and how tiers interact, as all also outstanding questions with TBD answers.
Physicalism, which is the most common worldview among philosophers today, offers an alternative approach, where many of these questions are assumed to not have consequential answers. Physicalism therefore presumes that the range of these TBDs is highly constrained as to what their ultimate answers will be.
If we're talking about Plato's Forms, then first of all, sort of, we would have to reflect upon the Form of Knowledge itself. I say "sort of" owing to Plato having passages like this in his writings:
Now, that which imparts truth to the known and the power of knowing to the knower is what I would have you term the idea of good, and this you will deem to be the cause of science, and of truth in so far as the latter becomes the subject of knowledge; beautiful too, as are both truth and knowledge, you will be right in esteeming this other nature as more beautiful than either; and, as in the previous instance, light and sight may be truly said to be like the sun, and yet not to be the sun, so in this other sphere, science and truth may be deemed to be like the good, but not the good; the good has a place of honor yet higher.
Somehow, the Form of the Good is the highest object of knowledge, but also something like the source of the Form of Knowledge, indeed of all the other Forms. At any rate, if knowledge generally depends on there being Forms of Knowledge and the Good "above" even that, then our knowledge of the Forms is tantamount to our knowledge that knowledge in general exists, encoded into our natural understanding, awaiting the day of its "awakening"/anamnesis. In other words, the unbeliever in the transcendent Forms doesn't have the full concept of knowledge at all in the first (negative) place, so there's an attempt at an "argument from self-refutation" that can be made, here.
As for modern abstract objects, the question of truly knowing about them is one of the most "haunting" in contemporary philosophy, at least for those engaged with the debate over it. One could try a move like the above and say that, "Abstract objects exist," is somehow encoded into the concept of knowledge per se nota, but it doesn't sound entirely plausible. A more popular "recent" argument was the one from indispensability, although even so, only just enough abstract objects were "accepted" as were needed for the background science to philosophically "go through." However, that argument has come under more and more fire over the years, alongside its partner-in-crime, the doctrine of ontological commitment more broadly.
So I would offer this consideration (not a solution, but an outline of a program for trying to find a solution): it is said that abstracta are "outside space and time" (whatever that's supposed to mean). However, they are also still reflected by particulars in spacetime, including spacetime itself (as we know it) as a particular. So aren't there abstract properties spatial and temporal? It is not usually thought that the abstract property of redness, for example, is itself red. A general triangle is not itself triangular (on pain of having to be any number of elementary, but not identical or necessarily even congruent, triangles in actual shape). So the pure properties of spatiality and temporality might not be spatial or temporal themselves. Still, could they not form a "conduit" or "link" or "portal" between the "world" of abstract objects, and our particular spacetime, seeing as our spacetime as a whole is instantiating them, using incredible amounts of energy and matter while doing so? Or, moreover, if there is a Form of Causation, so to say, and all caused knowledge in the empirical world instantiates the causation property, is there not another "doorway," here, between the "worlds"?
Addendum. Kant would not have been entirely fond of abstract objects, though he did countenance Forms in terms of deontic functions in the noumenal realm, albeit otherwise unknowable except by way of the responsibilities they engender in our phenomenal world. Still, in the second Critique, Kant does say something reminiscent of the above lines of emphasized reasoning:
Nothing worse could happen to these labours than that anyone should make the unexpected discovery that there neither is, nor can be, any a priori knowledge at all. But there is no danger of this. This would be the same thing as if one sought to prove by reason that there is no reason.