I think the best way to understand abstractions intuitively is to study lots of examples. To understand group actions, write some down. Then think about whether each is transitive, whether it's primitive, look at the stabilizers of elements. Verify that the theorem is true; try to understand why in each particular case. Look for examples where some hypothesis fails and see whether (and why) the conclusion fails.
That learning strategy reproduces (in part) the explorations that led to the useful abstractions of group theory. Mathematicians studying the symmetries of various geometrical objects realized that they could reason about the symmetry of just about anything by inventing an abstract language whose definitions captured the essence of the properties of symmetries.
Unfortunately, often the abstractions - the definitions and theorems - take center stage in teaching and learning. Students find it hard to understand what's going on without the examples. I would tell my students that for me the category of groups consisted of those groups with which I was personally familiar. I assigned homework that called for working out examples at least as much as proving theorems.
In your own discipline (computer science) there are analogous historical trends leading from examples to abstractions: the concepts of object oriented or functional programming languages, the development of abstract tools to reason about databases.