To rephrase the definition, for each $p\in S$, we have a neighborhood $U$ of $p$ and a coordinate chart $U\to V\subset \mathbb{R}^n$ of $N$ such that $S\cap U$ is the inverse image of a the intersection of $V$ with a $k$-dimensional linear subspace of $\mathbb{R}^n$ (in particular, a subspace of the form $(\star,\star,\star,\cdots, 0,0,0)$).
Locally, things do look like the $xy$-plane embedded in $\mathbb{R}^3$, defined by the vanishing of the $z$ coordinate. Of course, this won't hold for every chart, but we can cover $S$ with charts of $N$ such that, locally, a chart will determine what the points of $S$ are. If you have a point, and you have a chart, that tells you what the other points of $S$ are near $p$.
By the implicit function theorem, if $S$ is any submanifold of $N$, and $p\in S$, we can find a coordinate chart such that near $p$ the inclusion of $S$ into $N$ looks like the inclusion if $\mathbb{R}^k\subset \mathbb{R}^n$. Phrased differently, the condition that you wanted to hold happens for EVERY submanifold. So what this definition does is gives a topological restriction that you can't have different parts of the submanifold coming too close together.
For example, if you consider a slight modification of the topologist's sine curve, where you include a segment $(-1,1)$ of the $y$-axis and we connect that up with a path to the right hand side of the curve, you can view it as the image of the interval $(0,1)$ into $\mathbb{R}^2$ under a smooth, injective map, and so it is a submanifold of sorts. However, it is not a regular submanifold. Indeed, the induced topology (as a subset of $\mathbb{R}^2$ is different than the usual topology on $(-1,1)$). This is what the definition is meant to prevent, I believe.
Tu considers the smooth map $F : \mathbb R^3 \to \mathbb R^3, F(x,y,z) = (f(x,y,z),y,z)$ and shows that on some $U_p$ it restricts to a chart $F \mid_{U_p} : U_p \to F(U_p)$ on $\mathbb R^3$. In this special case this means nothing else than that $F \mid_{U_p}$ is diffeomorphism between open subsets of $\mathbb R^3$. The three coordinate functions of $F$ are $F_1= f$, $F_2 = p_2$, $F_3 = p_3$, where $p_i : \mathbb R^3 \to \mathbb R,p_i(x_1,x_2,x_3) = x_i$, is the projection onto the $i$-th coordinate.
But now by definition of $f$ we have have $S^2 = f^{-1}(0)$, i.e. $S^2 = \{(x,y,z) \in \mathbb R^3 \mid f(x,y,z) = 0 \}$. Therefore
$$U_p \cap S^2 = \{(x,y,z) \in U_p \mid f(x,y,z) = 0 \} = \{(x,y,z) \in U_p \mid F_1(x,y,z) = 0\} = \{(x,y,z) \in U_p \mid F(x,y,z) \in \{ 0 \} \times \mathbb R^2\} = (F \mid_{U_p})^{-1}(\{ 0 \} \times \mathbb R^2) .$$
This means that $U_p \cap S^2$ is the set of points $\xi \in U_p$ where the first coordinate of $F(\xi)$ vanishes, i.e. where $f(\xi)$ vanishes. This is what Tu expresses in the form
In this chart, the set $U_p \cap S^2$ is defined by the vanishing of the first coordinate $f$.
Note that this shows that (cf. Definition 9.1) $S^2 \subset \mathbb R^3$ is a regular submanifold of dimension $2$ because for each $p \in S^2$ we found the chart $(U_p,F \mid_{U_p})$ in the maximal atlas of $\mathbb R^3$ such that $U_p \cap S^2$ is defined by the vanishing of $n-2 = 1$ of the coordinate functions of $F \mid_{U_p}$.
Best Answer
Vanishing is sometimes a synonym for something being zero. The zero set of a function is where the function vanishes/goes to zero. Reading the next line you can see how the adapted chart is just where we have turned all coordinates greater than k to 0, in a sense the information in those higher coordinates has vanished.