This is a bit defeatist. Parsing the definition in your head is only the first level of understanding you can have about a mathematical structure. You don't really understand something until you can reinvent it (and in particular give a plausible answer to "why these axioms and not others?")
For example, to motivate groups, you could introduce the concept of a symmetry as a mapping from an object to itself that doesn't change its properties, introduce the idea of an isomorphism as a mapping with an inverse (where f and g being inverse means they compose to identity maps), put them together and postulate that a reasonable formalization of a symmetry is an automorphism (isomorphism from an object to itself), note that isomorphisms are closed under composition, and arrive at the definition of a group by considering sets of symmetries closed under finite composition (thinking of identity maps as the composition of 0 morphisms).
I'm sure there's a similarly conceptual way to motivate monads in functional programming. Hyland and Power have papers on algebraic theories of "effectful" operations and how they give rise to (finitary) monads, as one starting point.
It's a foundation of mathematics whose native objects are structures fundamental to higher mathematics (∞-groupoids, which are more or less equivalent to topological spaces up to homotopy), rather than awkward encodings (ZFC "sets", which are tree-like things), and where everything is automatically invariant with respect to abstract notions of equivalence (think isomorphism vs. equality).
The point is that it makes it much more realistic to check complicated math using computers, which will become more important as math inevitably gets more complicated.
Oh, and it happens to be a functional programming language. Which says something about functional programming.
Essentially the same way we reason about infinite mathematical structures in our heads: we define infinite structures by a finite system of generators and prove things by induction. The computer verifies that your inductive proof is correct.
In the fall of 1958, Zariski invited Grothendieck to visit Harvard. He was pleased to go, but made clear to Zariski that he refused to sign the pledge not to work to overthrow the American government which was necessary at that time to obtain a visa. Zariski warned him that he might find himself in prison; Grothendieck, perhaps mindful of the impressive amount of French mathematics done in prisons (think of Galois, Weil, Leray...) responded that that would be fine, as long as he could have books and students could visit.
I'm sat here, in my dressing gown. My fingers are greasy from eating crisps all morning. My back aches because I spent all night playing a stupid video game. The gin hangover isn't helping either. My week off work has been a wash out. I didn't write any code, I didn't cook anything other than pizza, and I'm beginning to smell of used dish water.
For example, to motivate groups, you could introduce the concept of a symmetry as a mapping from an object to itself that doesn't change its properties, introduce the idea of an isomorphism as a mapping with an inverse (where f and g being inverse means they compose to identity maps), put them together and postulate that a reasonable formalization of a symmetry is an automorphism (isomorphism from an object to itself), note that isomorphisms are closed under composition, and arrive at the definition of a group by considering sets of symmetries closed under finite composition (thinking of identity maps as the composition of 0 morphisms).
I'm sure there's a similarly conceptual way to motivate monads in functional programming. Hyland and Power have papers on algebraic theories of "effectful" operations and how they give rise to (finitary) monads, as one starting point.