To put it as concisely as possible, I would say that when you hear about 'non-realism' in Quantum Mechanics, it's not to be understood as a statement that NOTHING is real - merely a statement that the things we're used to thinking of as real are not real, or at least not as tangibly solidly real as we intuitively think.
Now to put it less concisely:
The reality underlying quantum mechanics is still in debate - there are so many interpretations, each with a different vision of what's really happening - and some of them keep what I call "naive realism" in tact, and others replace naive realism by giving reality to other constructs instead.
Naive realism is just simply the idea that, say, you throw a ball and look away and the ball continues to exist in a definite place and traveling at a definite velocity at every moment in time, even if no one is looking at it. This is also referred to in physics as "classical mechanics" or "classical physics".
So one way of conceptualizing non-realism in QM (which I wouldn't actually call anti-realism, I think that's opening it up to being confused with the metaphysical philosophical position of anti-realism) is, instead of an Electron existing at a definite place and flying at a definite velocity at every moment in time, even when unmeasured, instead think of it as The Universe is calculating all different trajectories and states that Electron could be in. The Universe is calculating what it would be like if it went this way, and if it went that way, and that way -- this is the "superposition".
The question, at the end of the day, that anti-realism poses isn't "is anything real?", it's "what things are real?" Because even if "the electron is at this place traveling at this velocity" isn't real, there's still something about the electron that's real even between times of being measured.