The axiomatic method involves replacing a coherent body of propositions (i.e. a mathematical theory) by a simpler collection of propositions (i.e. axioms). The axioms are designed so that the original body of propositions can be deduced from the axioms.
The axiomatic method, brought to the extreme, results in logicism. In their book Principia Mathematica, Alfred North Whitehead and Bertrand Russell attempted to show that all mathematical theory could be reduced to some collection of axioms. More generally, the reduction of a body of propositions to a particular collection of axioms belies the mathematician's research program. This was very prominent in the mathematics of the twentieth century, in particular in subjects based around homological algebra.
The explication of the particular axioms used in a theory can help to clarify a suitable level of abstraction that the mathematician would like to work with. For example, mathematicians opted that rings need not be commutative, which differed from Emmy Noether's original formulation. Mathematics decided to consider topological spaces more generally without the separation axiom which Felix Hausdorff originally formulated.
The Zermelo-Fraenkel axioms, the result of the axiomatic method applied to set theory, allowed the proper formulation of set theory problems and helped to avoid the paradoxes of naïve set theory. One such problem was the Continuum hypothesis.
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