Of course, if the distinction between realism and anti-realism is to be worth anything, it has to actually make sense on a basic level and be a view that people actually could conceivably hold. Understanding the view of anti-realism can give insight into realism.
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The disagreement between these two views is about ontology; it's a disagreement about what exists. Sometimes people use realism to signify the view that certain things exist and that this constitutes an explanation for certain phenomena. This is wrong: realism does not by itself provide an explanation. Much more is needed for an actual explanation than just what exists in an experimental situation. Especially in physics: one could express this by saying that the math has to work out as well. This is a blunder committed by some advocates of the Many-worlds interpretation. The interpretation is presented as a fix for quantum weirdness simply by supplying the right ontology, but this is misguided. For the many-worlds interpretation to be true, it must work out in the details, not just its ontological commitments.
Ontology, by itself, is not an explanation; theory is.
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That being said, ontology is a very robust way to divide and categorize different theories. Every theory admits of a certain "cast of characters" that it uses in theorizing. For example, classical mechanics has equations like "F=ma", which use the terms "force", "mass", and "acceleration". These terms can be used across different equations, and so long as the circumstances are the same, they can be used interchangeably in different theoretical equations. The "m" in "F=ma" and "p=mv" both stand for mass, and can refer to the same thing in the right circumstances.
Since this "cast of characters" of abstract terms are what the theory uses to describe a given experimental setup, they form the vocabulary that we use to describe the world in that theory. Different theories may have some abstract terms in common (like Newtonian and Quantum mechanics both use Hamiltonians), but they will still have abstract terms that do not overlap. This is a good way to show that two different forms of theorizing truly are parts of separate theories: their "cast of characters" are not the same.
Since different theories use different vocabularies to describe the world, this implies that they can admit different ontologies. That is to say, different theories describe different ontologies because they describe the world in different ways. The same can be said of philosophical interpretations of a physical theory. For this reason, ontology is a good topic to use to distinguish different interpretations of quantum mechanics.
This should be understood with two grains of salt, however: 1. ontology is not always the best way to distinguish interpretations and 2. not all interpretations disagree on ontologies, yet I would be inclined to say they are different theories (for example, some interpretations disagree on empirically-verifiable grounds.
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So what would an opponent of realism have to say that would make sense? They'd say that while of course we have the impression of real persisting objects in the world around us, these impressions (or certain ones) are best explained in terms of some mechanism other than real existing objects.
For example, it's possible all of the impressions of physically existing things we perceive come from some extra-physical being called God, and no physical thing actually exists. (This is roughly a view that the famous philosopher Berkeley argued for).
Or maybe we are simply brains in vats, hooked up to some elaborate machinery that gives us the impressions that we are in the physical world? (This is probably a much more palatable version to contemporary readers.)
Or maybe we are simply brains in vats, hooked up to some elaborate machinery that gives us the impressions that we are in the physical world? (This is probably a much more palatable version to contemporary readers.)
In both of these examples, our perceptions are not caused by what we would naturally think they are caused by. Instead, there is another explanation, and this explanation pictures reality differently. Anti-realism is a claim that what we perceive is best explained by an idea which does not admit of the existence of some thing.
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There's something else we can say about this alternate explanation. Not only is it asserted to be a better explanation, it is purported to be a better explanation of our impressions. If the anti-realist explanation is true, then it will bridge the gap between the impression as experienced and the explanation of that impression. This results in the meaningfulness of the words we use to describe the reality being replaced being undermined in a special way. If this anti-realist explanation is true, we would be able to exactly define the meaning of the words we used to describe the now-replaced ontology, and we'd be able to exactly define them in terms of how we came to know them. The meanings of the words we used would be definable through their epistemology.
That is far from clear, so imagine this example: there is someone permanently plugged into a realistic simulation of the outside world who was aware that this was the case. This person wanted to study how this simulation worked (and let's assume that he/she could genuinely understand it). Let's call this interested mind the 'matrix scientist'. Presumably, this matrix scientist would be able to figure out, in ideal circumstances, what sorts of mechanisms in the simulation were in action when he/she observed something, even though he/she was never taken out of the realistic simulation.
So let's say that this matrix scientist was trying to understand his/her own sense of smell, and the matrix produces this sensation through two distinct programs. You could even imagine this is a hardware difference, where these two aspects of smell run on two physically separate machines. The important point is that there are two distinct causal processes for producing the scientist's sense of smell. Let's call them process A and process B.
The matrix scientist could study these different processes and learn their differences, their similarities, how they work together, etc. Even after learning all about them, however, that textbook knowledge he/she had would still not count, strictly speaking, as knowledge of the unreality of his/her sense of smell. After all, that knowledge has not yet been connected to his/her perception of smell. In order to do that, the matrix scientist would have to be able to describe his/her sense experiences of smell in terms of these processes. Suppose process A controlled the smell of crayons, and everything else was controlled by process B. The matrix scientist might at some point smell crayons, and because he/she knows that this is a result of process A, he/she could say "process A is occurring" just as easily as he/she might say "I smell crayons". The matrix scientist could go on to do this with all aspects of his senses, given enough information about the causal forces that give rise to his/her sensations. The end result of this is a language which can be used to describe all aspects of the reality of perception without using any sensory concepts like "smell".
What this elaborate example is meant to illustrate is that, in order for some natural notion of reality (like sensory reality) to be replaced with another notion of what is real, we have to have ways of describing our old reality in terms of the new.
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So at the end of the day, anti-realism is a commitment to the following:
1. There may be some realist explanation of our impressions, but the anti-realist thinks that there is a better explanation that does not use the disliked ontological concept.
2. Our sense impressions can be completely described in terms of this alternate causal process.
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