A hypoallergenic something is something that is unlikely to cause an allergic reaction. Therefore, a hypoallergenic eyeshadow brush is one that is unlikely to cause allergic reaction when you use it around your eyes. An allergic reaction happens when your immune system is trying to fight off a certain substance. The substance could be ultimately harmless, but your immune system doesn't know that. A skin allergic reaction can cause a burning itchy sensation. It can cause a rash or swelling, or both. These are not something you want to experience with your face.
Now, you might be wondering, why the focus on eyeshadow brush? Well, you want all your makeup products to be hypoallernegic. However, a special focus must be directed towards eyeshadow brushes because the skin around the eyes are very sensitive. They are significantly more sensitive than other parts of your face. That is why many beauty products, especially those chemical exfoliants, are advised not to be applied on the skin around the eyes and mouth. An allergic reaction becomes so much worse when it happens around your eyes.
So what makes an eyeshadow brush hypoallergenic? It comes down to the type of materials and processes used to manufacture and package the brush. There are so many types of substances that can cause allergic reactions, that listing them one by one is impossible. So when doing your shopping, you want to make sure that you search or ask for a hypoallergenic product. However, you cannot just believe what the manufacturer claims. You want certification from government or third-party testers that their products are truly hypoallergenic.