Does reality actually exist? If so, which things are part of reality, and which things are not? There are any number of philosophical tarpits that bottom out in a question like this. Does infinity exist? Does consciousness? Does love? Irrealism is a philosophical stance which rejects the premises of these questions.
The fundamental claim of irrealism is this: the only way we interact with anything at all is through sense and symbol. Sense is the raw data that comes from the world around us; symbols are our representations of that data in our minds. That is all we can say about "reality"—that we collect sense data, and we think about that sense data symbolically.
This idea caches out in various ways, according to who is using it. In The Matrix, humans are fed synthetic sense data by machines, which places them in a virtual world that has nothing to do with the real one. The thought-experiment version is "how do you know that you're not just a brain in a jar?" You can't prove it, because anything you can point to might be an illusion. It's all just sense data.
It's important to note that this is entirely a negative claim. There's no second step where we go "therefore we are in the Matrix". There's no material or social consequence involved; if you're actually a brain-in-a-jar, that doesn't affect your life one bit. So why care?
To me, irrealism is about epistemic humility. I believe that we are too quick to forget that each of us occupies a limited, incomplete, and frequently erroneous viewpoint in the world. Contemplating the inherent limitations of knowledge—the fact that we can never see "through" the sense data to bare reality—helps to keep one's overconfidence in check.