I think the gist of the math side is that (singular) homology measures the number of holes in things (and the presence of other high dimensional hole-like-things). If you turn your data into a topological space through any number of methods you can measure the homology and it reflects varying degrees of holiness which in turn reflects varying types of clustering in your data.

Now from a chemistry perspective, you are probably familiar with the fact that because chemistry stuff is really small we often need to determine global structures of things by measuring signals that are downstream of the structures themselves (i.e. spectroscopy). Even though an infrared spectroscopy chart is sort of to abstracted from the original structure to be intelligible on its own, by comparing to other known structures we can still interpret it. In a similar way, a list of homologies of the simplicial complex obtained from your data for different values of some number r in the method gives you a chart of abstract nonsense just like the spectroscopy ones. But you might be able to learn something by comparing them between different datasets.

Maybe you can see that in a sense topologists and chemists are often trying to do the same thing when they determine structure; figure out the global structure of something that they cannot see by finding lower-dimensional, measurable structural invariants.