In Lee's Introduction to Smooth Manifolds he gives the following definition:
Given a rank-$k$ distribution $D \subset TM$, let us say that a smooth coordinate chart $(U,\varphi)$ on $M$ is flat for $D$ if $\varphi(U)$ is a cube in $\mathbb{R}^n$, and at points of $U$, $D$ is spanned by the first $k$ coordinate vector fields $\partial/\partial x^1, \ldots, \partial/\partial x^k$. In any such chart, each slice of the form $x^{k+1} = c^{k+1}, \ldots, x^n = c^n$ for constants $c^{k+1}, \ldots, c^n$ is an integral manifold of $D$.
I would like to understand the intuition behind these objects as they seem important. If possible, I would also like to have a mental picture of what they represent and they are called flat. Lee provides a picture in his book (Fig. 19.2 on page 496) but I don't fully understand what it represents.
Why do want $\varphi(U)$ to be a cube? How does the condition that the first $k$ coordinate vector fields $\partial/\partial x^1, \ldots, \partial/\partial x^k$ span $D$ at each point relate to being "flat"? Lastly, why does the condition that the last $n-k$ coordinates being equal to a constant imply we have an integral manifold of $D$?