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.