The following is the start of the proof of proposition 9.5 in Hartshorne's chapter III. The setup is this: $f: X \to Y$ is a flat morphism of schemes of finite type over a field $k$, $x \in X$ and $y \in Y$, and we define $\dim_P S := \dim \mathcal{O}_{P, S}$ for any scheme $S$. In the proposition, we compare the numbers $\dim_x(X_y)$, $\dim_x X$, and $\dim_y Y$.
First we make a base change $Y' \to Y$ where $Y' = \operatorname{Spec} \mathcal{O}_{y, Y}$, and consider the new morphism $f': X' \to Y'$ where $X' = X \times_Y Y'$. Then, $f$ is also flat by (9.2), $x$ lifts to $X'$, and the three numbers in question are the same.
The last sentence is where the main gap is, as we have to check all the numbers are the same and that $x$ does indeed lift to $X'$. This previous answer to a question explains well why $x$ lifts to $X'$ so I'm only worried about the numbers.
Now, $y \in \operatorname{Spec} \mathcal{O}_{y, Y}$ is a closed point so that $\dim_y Y' = \dim \mathcal{O}_{y, Y} = \dim_y Y$, so this number certainly stays the same.
For the other two, I'm somewhat at a loss. The best I could do is figure out that you can reduce to the case that $X$ and $Y$ are affine so that this is a statement about commutative algebra. This wasn't very fruitful, and I wonder if there is a more global topological way to do this more clearly.
Thanks for any help!
EDIT: This really does come down to understanding the local situation. I found this post and KReiser's nice answer to be incredibly helpful.
Best Answer
I will prove one of the statements.
All statements are local, so let $X=\mathrm{Spec}\ A$ and $Y=\mathrm{Spec}\ B$, for finite-type $k$-algebras $A$ and $B$, with a morphism $\varphi\colon B\to A$. Let $x=\mathfrak p\in X$, so that $Y'=\mathrm{Spec}\ B_{\varphi^{-1}\mathfrak p}$ and $X'=\mathrm{Spec}(A\otimes_BB_{\varphi^{-1}\mathfrak p})$.
We want: $$\dim((A\otimes_BB_{\varphi^{-1}\mathfrak p})_\mathfrak p)=\dim(A_\mathfrak p).$$ This follows from the isomorphism $$\begin{align*} (A\otimes_BB_{\varphi^{-1}\mathfrak p})\otimes_AA_{\mathfrak p}&\cong A_{\mathfrak p}\\ (a_1\otimes b)\otimes a_2/s&\mapsto a_1a_2\varphi(b)/s\\ 1\otimes 1\otimes a/1&\leftarrow a. \end{align*}$$