Intuition of: Every proper subspace of a normed vector space has empty interior?
Since there's a proof for it, then it's true, but I have troubles understanding the intuition of a "filled space" having every subspace with empty interiors.
Does it mean that the structure of normed spaces is somehow "discrete"? That it has holes in it?
The intuition here is that if $V$ is a normed vector space, and $B$ is some open ball around the origin of $V$, then $B$ "contains every direction" in $V$. Indeed, if $v \in V$ is any vector, then I can always pick $\epsilon$ small enough such that $\epsilon v \in B$. So if $U \subseteq V$ is a proper subspace, then every open ball $B$ around the origin has points not belonging to $U$, so cannot be an interior point.
Imagine any proper subspace in $\mathbb{R}^3$, so a point, a line, or a plane. It is impossible to put a solid ball, however small, properly inside that subspace. There are no holes in the subspaces, it's just that the ball you are using still comes from the ambient space $\mathbb{R}^3$.