What is unknown in Bellman Equation? \begin{align}
 V(W)=\max\limits_{W'\in[0,W]}\qquad&amp; u(W-W')+\beta V(W')\qquad\forall W
\end{align}

$\textbf{My Question}$: Why is the unknown in the Bellman equation $V(W)$ itself? Isn't the choice variable tomorrow's state, $W'$?
The original problem was probably of the form $$\max_{\{W_t\}_{t=1}^\infty}\sum_{t=0}^\infty \beta^t u(W_t-W_{t+1}),$$
$$\mbox{s.t. } W_{t+1}\in[0,W_t] \ \forall \ t,  ~~ W_0 \mbox{ given}$$
When formulated that way there is no need to solve for any function $V$, but the infinite elements of the sequence ${\{W_t\}}_{t=1}^\infty$ are all choice variables. Bellman's main insight is that (under some conditions) the problem can be simplified. Instead of solving for an infinite number of variables, you could solve for a single "policy function" that tells you what to do each period as a function of the current state variables, and a "value function" that summarizes what is the value (in terms of today) of inducing each possible state in the future. Even if there is no uncertainty, it is clear that this value function is an endogenous object, since it must somehow capture the future stream of utility as a single function of the future state. In your case, that function is  $V(W')$. This is why the Bellman equation: 
$$V(W)=\max_{W'\in[0,W]} u(W-W')+\beta V(W')$$
has two properties: 

1) it lacks the subscripts on the $W$'s, this reflects that you will solve for $W'$ as a function of $W$, i.e. the policy function. 

2) has a new object, $V$, that was not in the original problem and it is a function of $W$ (or $W'$) and just by eyeballing the two problems you can see that for them to be equivalent $V$ must be the discounted sum of future payoffs assuming future actions are also optimal. So it is not a choice variable, but definitely, an endogenous object that you need to find when solving the problem.

You can see that the problem of optimally choosing infinitely many objects is transformed into one of choosing a finite number (2) of functions, which are infinite-dimensional objects. Brilliant!