Why can fix $W_0 \subseteq \kappa$ and select $W$ such that $W_0\subseteq W \subseteq \kappa$? Lemma V$.2.19$ (book Kunen)
In $M$, let $\mathbb{P}=Fn(\kappa,\omega)$, where $\kappa$ is any cardinal. Let $K$ be $\mathbb{P}$-generic over $M$. Then $M[K]\models \mathfrak{d} \geq \kappa$
Proof:
If these result is false, then, working in $M[K]$, there would be a cardinal $\theta<\kappa$ and a function $\overrightarrow{h}:\theta\times{\omega }\to \omega $ such that $\{h_\alpha: \alpha< \theta\}$is a dominating family, where $h_\alpha(n)=\overrightarrow{h}(\alpha,n)$. Of course, $\theta \geq \omega^M_1=\omega^{M[K]}_1$.  Let  $\tau \in{M^{ \mathbb{P}}}$ be a nice name for a subset of $( \theta\times{\omega })\times{\omega }$ such that $\tau_K=\overrightarrow{h}$.

Working in $M$,  fix $W_0\subseteq \kappa$ such that $|W_0|\leq \kappa $and all forcing 
  conditions mencioned in $\tau$ come from $Fn(W_0,\omega)$; then choose $W$ with $W_0\subseteq W \subseteq \kappa$ and $|\kappa\setminus W|=\aleph_0$. Note that $\tau$ is a $Fn(W,\omega)$-name and $Fn(\kappa\setminus W,\omega)\cong Fn(\omega,\omega)$.

Why can fix $W_0 \subseteq \kappa$ and select $W$ such that $W_0\subseteq W \subseteq \kappa$ ?
 A: First of all you meant to say $|W_0|\leq \theta$ I believe, otherwise it is just trivial by taking everything. 
For the first question, $\tau$ as a nice name for a subset of $(\theta\times \omega)\times \omega$ (to be more precise, this is a canonical name in M) can be represented as $\{(\check{\delta},A_\delta): \delta\in (\theta\times \omega)\times \omega\}$ where $\check{\delta}$ is a canonical name for $\delta\in (\theta\times \omega)\times \omega$ and $A_\delta$ is an anti chain in $Fn(\kappa, \omega)$. So essentially $W_0$ is the union of all domains of the conditions that appear in these anti chains. More precisely,
$W_0 = \bigcup_{\delta\in (\theta\times \omega)\times \omega, p \in A_\delta} dom(p)$. As the forcing notion is c.c.c, $|W_0|\leq \theta$. 
For the second question, as $|W_0|\leq \theta < \kappa$, $\kappa \backslash W_0$ is of cardinality $\kappa$. As M is a ZFC model, you might well-order $\kappa \backslash W_0$ and leave out the first $\omega$ elements, i.e. list $\kappa \backslash W_0$ as $\langle a_i: i<\kappa\rangle$. $W=W_0 \cup \{a_i: i\geq \omega\}$.
