Broadly, we just want to understand how different constructions 'interlock', which can be made rigorous by a wide variety of very controlled methods. In alchemical metaphor: we want to purify solutions to problems, distill and classify structural species (the 'results' of 'performing constructions'), coagulating species into useful combinations, et cetera. I find this colorful language useful because mathematicians habitually appropriate ordinary words.
All sciences seek understanding, including category theory. The differences are measured in the scope of their applications. Category theory has been carried forward by the explicit intention to expand its scope, perhaps to include ALL other scopes. My idea of "scope" is similar to the idea of "context" in dependent type theory (which is roughly "cartesian closed category theory" via Lambek-Curry-Howard). Contextuality is important in logic as well as quantum phenomenology. For example, it's worth wondering about the scope of application of a phenomenological physical theory. "Where/when" and "how", exactly, does a description "break down"? When does a logic fail, creating demand for new logics to take its place? Is there a "best" or "right" way to go about doing math, logic, science, and engineering? Many category theorists seek beyond the binaries and Booleans of "classical foundations", based on "material set theory". Rather than settling for negative answers to existence questions, we refine to questions concerning *obstructions* to existence. Can we classify the obstructions? There may be a variety of substantially different obstructions, some being essential and others capable of being "deformed away" with some structural mutation. These lines of inquiry have brought awareness of deeper and richer connections between structures. We discover and invent new species of information, and adapt our worldview to accommodate the expanding scope.
Exploring the mathematical polycosm feels like hacking through a dense jungle of tangled ideas. The tools of expedition are formal constructions, expressed in formal languages. These languages evolve and adapt to the shifting demands of application, splintering into multitudes and giving birth to new languages capable of differentiating the previous generations (which were born in order to differentiate previous generations of languages, et cetera). Category theory engineers tools for efficient expression and manipulation of linguistic constructions, and that pretty much sums up my motives for studying and applying it.