tl;dr version: Kant and Hume both recognize causality isn't out there in the world. Both accept the mind does it. Kant and Hume both don't see how we can perceive anything without it.
For Kant, this is a feature. Kant thinks its a feature of our reason. For Hume, it's a bug that give us the illusion of a world that makes better sense than it does. If you think being able to makes sense of the world is truth-revealing, then you agree with Kant. If you think it is just feeding our illusions, then you agree with Hume. That's what Kant's "refutation" hinges on.
Kant criticizes both empiricists and rationalists in his Critique of Pure Reason (hereafter CPR). The famous quote line regarding this is here:
Thoughts without intuitions are empty, intuitions without concepts are blind. (CPR B 75).
In short, his criticism of empiricists is that they are not at the end of the day empiricists about what it means to know. Roughly sketched, empiricism is the view that we perceive with our senses is true as a counterpart to Descartes' clear and distinct ideas. The problem is that there are two related questions:
(1) What is true?
(2) How do we know it?
The quote is also about this and compactly expresses Kant's view. But let's start with Descartes.
For Descartes, clear and distinct ideas are true, because of God and the fact that as long as I exist, I exist. For Locke, primary qualia are true. We know them through secondary qualia. Those two strands get upgraded by subsequent rationalists and empiricists. So Spinoza and Leibniz supply a more robust logic to how reason gets us truth and how we might possibly know it. For both we wind up with a kind of ration-infused universe that moves according to the laws of logic (and our sensory perceptions remain dubious). On the empiricist side, Berkeley turns out to be an idealist. Hume, in turn, sweeps out ideas like cause and effect (only to restore them as mental ideas of habit).
Kant's critique is a return to these problems via Hume's answer. Kant agrees with Hume that the rationalists are wrong about what is real. For him, what is real are empirical things. But where Kant disagrees (or possibly disagrees depending on how you interpret both when they talk about this) is on how we arrive at knowledge. Or at least how we are to view the mechanism.
Kant and Hume both agree that we've got causality in our minds and a few other notions and these help us sort the world. For Hume, these ideas come from experience but cannot be trusted because they are generalizations we engage in. For Kant, that won't do. One of Kant's argument for this point is that we cannot arrive at causation without a category of cause and effect already in our minds. In other words, he thinks the Humean skeptical project is inaccurate as a description of mind.
For Kant, our minds already and apply categories of understanding and forms of sensibility to to the real things they encounter. Thus, Kant's answer to how we know is that as rational beings, we frame our experience in the unity of apperception using categories and forms. For him, this is knowledge (at least of the human sort) -- which is part of why he thinks we cannot know things in themselves.
One major difficulty that makes it hard to answer is that it's not clear how different Hume and Kant really are on this view. The problem is that the a priori notion of cause that Kant talks about is one that he distinguishes from empirical experiences of causal relations. And then it's not clear how one moves between the two. Because Kant denies that you can arrive at the a priori sense of cause by induction.
That's the best I can really do with such a broad question...