How do reproductive barriers relate to the biological species concept?
The biological species concept relies on the idea a group differentiates into separate species when they can't mate
Species change from adaptation, and this is more so when an adaptation is favored. Evolution and adaptation posits that over time a group from one species can adapt so much they become a different species. In effort to classify groups you could say that a species A of felines and species B of felines must be totally different because although they came from a common ancestor, they can't mate. This is different than other methods of defining species, like RNA differences.
Note that reproductive barriers can be physiological, like species A evolving to be taller so mating isn't physically possible. Barriers can also be due to habitat differences. Birds may live in separate parts of the forest, and that may reduce mating competition, but the two groups may still be physically compatible; so the theory isn't an explanation for everything.