Baire's theorem is considered fundamental for functional analysis. It could be simply stated as
In complete metric spaces (e.g.) countable intersections of dense open sets are dense.
But in many textbooks, there is instead a definition of what it means for a set to be of first category (meager), of second category, or residual (comeager). Then a Baire space is defined as a space in which every residual set is dense, and Baire's theorem becomes "complete metric spaces are Baire spaces".
The terms "first category", "second category", etc. are not needed for the statement of Baire's theorem, nor do they provide any intuition for it.
What is the bigger picture for which these concepts are needed?
a meagre set (...) is a set that (...) is in a precise sense small or negligible
In what sense?