Intersection Theory flat/proper commutation

Question

I have a problem with proposition 1.7 from Fulton's book, "Intersection Theory", that states that if $f$ is a proper morphism between $k$-schemes, $X$ and $Y$, and if we make a flat base change $g: Y'\to Y$, then $g^\ast f_\ast=f'_\ast g'^\ast$, where g' and f' are the maps that complete the fiber square. I really dont understand the proof, nor the reduction to the commutative algebra neither the commutative algebra per say. I'm really confused about this, so could somebody explain?