I'll give an excerpt from the book:
As previously, we start by constructing the blow up of a disk along a coordinate plane. Let $\Delta$ be an $n$-dimensional disk with holomorphic coordinates $z_1,\ldots, z_n$ and let $V\subset \Delta$ be the locus $z_{k+1}=\ldots = z_n = 0$. Let $[l_{k+1},\ldots , l_n ] $ be homogeneous coordinates on $\mathbb{P}^{n-k-1}$, and let $\widetilde{\Delta}\subset \Delta \times \mathbb{P}^{n-k-1}$ be the smooth variety defined by the relations: $$\widetilde{\Delta} = \{ (z,l) : z_i l_j = z_j l_i, k+1\le i,j \le n \}$$ The projection $\pi:\widetilde{\Delta} \to \Delta$ on the first factor is clearly an isomorphism away from $V$, while the inverse image of a point $z\in V$ is a projective space $\mathbb{P}^{n-k-1}$. The manifold $\widetilde{\Delta}$, together with the map $\pi: \widetilde{\Delta}\to \Delta$, is called the blow-up of $\Delta$ along $V$; the inverse image $E=\pi^{-1}(V)$ is called the exceptional divisor of the blow up. $\widetilde{\Delta}$ may be covered by coordinate patches: $U_j=(l_j\ne 0), \ j=k+1 ,\ldots, n$ with holomorphic coordinates
$$z_i = z_i , \quad i=1,\ldots , k, \\ z(j)_i = \frac {l_i} {l_j} = \frac {z_i} {z_j}, \quad i=k+1,\ldots, \hat{j} , \ldots , n, \\ z_j = z_j$$
on $U_j $.
What I don't understand and I suspect it's a typo is the last line: isn't $z_i=z_i, z_j=z_j$ always true regardless of the properties of holomorphic coordinates?
What that notation means is that you introduce new coordinates $(Z_1, \dots Z_n)$ defined as
$$Z_i = z_i , \quad i=1,\ldots , k, \\ Z_i = \frac {z_i} {z_j}, \quad i=k+1,\ldots, j-1, j+1, \ldots , n, \\ Z_j = z_j ,$$
i.e. you keep the old first $k$ coordinates and the $j$-th one, and you only alter the last $n - k$ according to the middle formula, taking care though to skip the $j$-th coordinate (which, as said, stays unchanged).
More clearly,
$$(Z_1, \dots, Z_n) = \left( z_1, \dots, z_k, \frac {z_{k+1}} {z_j}, \dots, \frac {z_{j-1}} {z_j}, z_j, \frac {z_{j+1}} {z_j}, \dots, \frac {z_n} {z_j} \right) .$$