Every irreducible component of the exceptional set has codimension one when scheme is normal and $\mathbb{Q}$-factorial.

Question

I want to calculate the dimension of the ring $\mathcal{O}_{X,x}$ in the figure below. But I can't understand his proof, for example, what does "$(u=v=0)$ has codimension two." mean?

Can this proof be transformed into an algebraic proof? (I guess that $\mathcal{O}_{X,x}$ dominates $\mathcal{O}_{Y,y}$, $t_1\in m_{X,x} \setminus \mathcal{O}_{Y,y}$ and $u,v\in m_{Y,y}$.)