In dimension of intersection of hyperplanes in the answer it is explained that the intersection of a affine hyperplane and $m$-dimensional affine subspace can have 3 scenarios.
The first two are pretty much logical and easy to understand, but why is the proper intersection exactly of dimension $m-1$ (one less than the $m$-dimensional affine subspace)?
To have a good notion of hyperplane I restrict myself to a $d$-dimensional vector space with $d<\infty$.
Suppose we have an affine hyperplane $H$ and another affine subspace $A$ of dimension $m$.
If $m=d$ we have $A=V$ so $\dim A\cap H = \dim H = d-1 =m-1$.
If $A$ is contained in $H$ then $\dim A \cap H = \dim A = m$.
If $A$ is parallel to a subspace of $H$ and they don’t share a common point, then $A\cap H=\emptyset$ and the dimension of the intersection is not defined.
Otherwise they do intersect and it suffices to restrict to the case that both $A$ and $H$ are linear subspaces (not affine anymore). We find $A+H=V$, since otherwise $A$ would be contained in $H$. Hence the dimension formula yields $$\begin{align*} d &= \dim V\\ &= \dim (A + H)\\ &= \dim A + \dim H - \dim (A\cap H)\\ &= d-1 + m - \dim A\cap H \end{align*}$$ Thus we get $\dim (A \cap H) = m-1$.
