ct.category theory - What are the advantages of phrasing results in terms of exact sequences and commutative diagrams?

If you are asking why very elementary results like the first isomorphism theorem are phrased
in the language of exact sequences/commutative diagrams (rather than why this language is used at all), then there are (at least) two answers: (1) for those who are used to using this language, they frequently think about even those elementary results in terms of it, and so it is natural to write them in that language; (2) we want to train students to learn this language, and have to start somewhere, so we begin by taking elementary results that can be understood in another way, such as the first isomorphism theorem, and then rewrite them in this language for pedagogical purposes.

If you are asking why people use this language at all (which is to say, why are there many people to whom (1) above applies, and why do we want to engage in the educational practice labelled (2) above), then Keith Conrad gives a pretty good answer.

At a slightly broader level of generality, one might cite the old saying "a picture is worth a thousand words", and note that a well-chosen diagram or exact sequence can convey a lot of mathematical information in a succinct and intuitive way (the intuition coming once you have some familiarity with this way of thinking). We have a lot of mathematics to remember, and are always looking for ways to compress our descriptions of things without losing information or becoming unclear. Well-chosen definitions and terminology are one way this is achieved; well-drawn diagrams and exact sequences are another.

Finally, one could note that contemplating diagrams appeals (however slightly) to geometric modes of reasoning. Typically, any method which allows one to import some kind of geometric reasoning into algebra is welcome, since it brings less typically algebraic ways of thinking to bear on algebraic problems.