Let's begin with the definition of immersion:
Definition: An immersion from an open set $U\subset \mathbb R^m$ to $\mathbb R^n$ is a differentiable function $f:U\to \mathbb R^n$ such that for every $x\in U$, the derivative $f'(x):\mathbb R^m\to \mathbb R^n$ is an injective linear transformation.
The book I'm reading states that we can use the theorem below to say if an immersion is sufficiently smooth, then it's possible to set new coordinates in the neighborhood of every point of the image so that $f$ is locally an inclusion.
Immersion theorem: Let $U\subset \mathbb R^m$ be an open set and $f:U\to \mathbb R^{m+n}$ be a strongly differentiable function at $a\in U$.
There are open sets $Z\subset \mathbb R^{m+n}$ and $V\times W\subset \mathbb R^m\times\mathbb R^n$ with $f(a)\in Z$ and $(a,0)\in V\times W$ and a homeomorphism $h:Z\to V\times W$ such that $hf(x)=(x,0)$ for every $x\in V$ and $h$ is strongly differentiable at the point $f(a)$.
If $f\in C^k (k\ge 1)$, it's possible to restrict if necessary $V$, $W$ and $Z$ such that $h$ is a $C^k$ diffeomorphism.
I didn't understand why this comment is true. For me we only have that there is another function whose the composition with the immersion is an inclusion.