The motivation behind axiomatisation

You might be familiar with the programming rule of abstraction that the third time you write a piece of code to do something, you should refactor your program to move that functionality out into its own class or function.

Commutative rings, division rings, and fields have come up way more than three times in mathematics. :)

Encapsulation is just as useful in mathematics as in programming to hide away extraneous information, thus making it easier to work with something. The best example of this sort of thing I know came up in quantum physics (which I will describe somewhat glibly): when the field began, all of the math was deep in the theory of partial differential equations and differential operators, which are quite complicated and rather laborious to work with.

Then Dirac came along and said "this is all just linear algebra", introduced Hilbert spaces and bra-ket notation, and poof quantum physics quickly became much, much easier to work with and reason about.

My history is somewhat weak, but I believe commutative algebra really originated with the development of algebraic number theory: when people started studying number rings and number fields in the attacks on Fermat's Last Theorem, they quickly realized that there was a lot of interesting structure there... as well as lack of structure (e.g. many number rings don't allow unique factorization into elements).

As new structure was discovered, new ways rings could be pathological, and new sorts of arguments for reasoning about rings, definitions were given that supported these sorts of structures and reasoning, or separated the rings we are actually interested in from the badly behaved ones.

e.g. the notion of a Noetherian ring came about when Emmy Noether realized the power of making arguments based on the ascending chain condition: an infinite chain of ideals I1⊆I2⊆⋯I1⊆I2⊆⋯ is eventually constant. Rings people are interested in often have this property, but general rings do not: the structure "Noetherian ring" is thus defined, so that people have words to say that they are working with rings that allow this sort of argument.