Given a symplectic manifold $(M,\omega)$ with integral symplectic form, that is $$\omega \in \text{Im}(H_2(M,\mathbb{Z}) \to H_2(M,\mathbb{R})),$$ one can form a so-called prequantization space, that is a $S^1$-bundle $$ \pi : (V, \alpha) \to (M,\omega),$$ where $\alpha$ is an $S^1$-invariant $1$-form on $V$ satisfying $$\pi^* \omega = d \alpha.$$ This makes $(V,\alpha)$ into a contact manifold.

What is the physical intuition behind this construction ? I know that it corresponds to the notion of geometric quantisation, but I have trouble seeing why $(V,\alpha)$ could represent a "quantum" space associated with $(M,\omega)$. For instance, what is the meaning of the fibres of $\pi$ ?