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To first give you some context, the thing on the top is from The “Representable Functors” chapter of Category Theory for Programmers. So technically, you only need to read 230 Pages of a maths textbook to get it ;)
But this isn’t exactly what you asked for, so I’ll try to help you get it as best I can with my limited understanding of the subject. First of all it would be helpful to know what your prior knowledge in Maths, especially Set theory, is?
From what I gathered from the Wikipedia page of Kripke Models, they take the form of something like a graph (nodes and edges). My mediocre knowledge of higher math makes me wave my hands and say something about Category theory also working with graph-like structures and that sheaves and topoi definitely come up there, too.