A coloring of a set $X$ by $Y$ is just a function $f:X\rightarrow Y$ The term coloring is used because the function can be thought of as assigning a ``color'' from $Y$ to each element of $X$

Any coloring provides a partition of $X$ for each $y\in Y$ $f^{-1}(y)$ the set of elements $x$ such that $f(x)=y$ is one element of the partition. Since $f$ is a function, the sets in the partition are disjoint, and since it is a total function, their union is $X$