The step I am confused on is in yellow background
Let $S$ be a subspace of $\mathbb{R}^n$ and $S^\perp$ be its orthogonal complement.
Let $\{\textbf{x}_1,\dots,\textbf{x}_r\}$ be a basis for $S$ and $\{\textbf{x}_{r+1},\dots,\textbf{x}_n\}$ be a basis for $S^\perp$
Suppose $$ c_1\textbf{x}_1+\dots + c_r\textbf{x}_r + c_{r+1}\textbf{x}_{r+1} + \dots c_n\textbf{x}_n =0 $$
Defining $\textbf{y} =c_1\textbf{x}_1+\dots + c_r\textbf{x}_r$ and $\textbf{z} = c_{r+1}\textbf{x}_{r+1} + \dots c_n\textbf{x}_n$ we have that
$$ \textbf{y}+\textbf{z} =0 $$ thus $$ \textbf{y}=-\textbf{z} $$
Thus $\textbf{y}$ and $\textbf{z}$ are both elements of $S\cap S^\perp$
I am confused on that last statement (in yellow)
Is the reasoning behind the step that $z\in S$ because it is $(-1) \textbf{y}$ and thus is a linear combination of the basis elements of $S$? (and similar reasoning for why $y\in S^\perp$)
Since subspaces are closed under addition and scalar multiplication, we have $y\in S$ and $z\in S^{\perp}$. So you're correct in saying that we also have $z\in S$ because $z=-y$ and $y\in S$ so $-y\in S$. Similarly for showing $y\in S^\perp$.