A physicist, a biologist, and a chemist are all trying to study a particular differential equation. Each of these differential equations are mathematically identical but full of specialized constants and terms related to each field. The physicist has an equation about mechanical stress and the biologist has an equation about the movement of cells. Each of the three scratch their heads: how can one solve such a difficult equation with dozens of components and factors?
They go to the mathematics dept. to speak to somebody who works on differential equations. When the mathematician sees these equations she is considerably less frightened. Because she does not know about cells or metabolic rates or chemical reactions, she just sees an equation full of arbitrary constants, and is not confused by the unnecessary information about what those constants mean. She recognizes all three problems as a certain sort of differential equation that she has studied in the past, and solves them quickly using the general theory she has developed.
In this analogy, physics, chemistry, and biology are different branches of math. Sometimes these fields have specific problems phrased in different languages which, on some intrinsic level, are logically identical. In certain situations this extra information only confuses us and is not necessary for solving the problem. When we show this problem to category theory, it does not see the specialized details but only the inherent underlying problem, and we can then apply the broad framework of category theory to tackle all these issues at once.