What is the result of the Bellman Equation
...the result of applying sup operator is a NUMBER...


Read it carefully. The equation is 

$$
v(x_0) = \sup_{ \{x_t \}_{t \geq 1}} \cdots \quad (1)
$$
This defines a function $v$, called the value function, which is a type of indirect utility function. For given $x = x_0$, the value of $v(x)$ is defined to be the sup on the RHS, taken over feasible sequences $\{ x_t \}$.


  ...there is a functional equation, the result is a function...


Yes, the function $v(\cdot)$ defined by $(1)$ satisfies the function (Bellman) equation. Equality is in the sense of functions. The LHS is $v(\cdot)$ defined by $(1)$. The RHS is another function 
$$
\nu(x) = \sup_{y\in\Gamma(x)}[F(x,y)+\beta v(y)].
$$ 
The claim---the dynamic programming principle---is that they are equal, $v(\cdot) = \nu(\cdot)$.