how it relates to the Fubini Theorem?
Let $\chi_A$ be the characteristic function of $A$: that is, $\chi_A(x)=1$ when $x\in A$ and $0$ otherwise. The integral of $\chi_A$ with respect to whatever measure is equal to the measure of $A$. This is why the characteristic function is used to make the transition from measures to integrals and back.
Fubini's theorem is about integrals. Applied to $\chi_A$, it says that the integral of $\chi_A$ can be found by integrating over each vertical slice first, and then integrating over $c$. Well, if every slice has measure zero, then the integral over every slice is zero. Then $\int 0\,dc =0$, and the conclusion is that $A$ has measure zero.
what does this mean?
Think of the calculus exercise about finding volume by integrating the area of cross-sections. If the area of each cross-section happened to be zero, the volume is zero. That's all there is to it, but a rigorous proof takes some effort, mostly to make sure that the concepts are properly defined.