In Dan Freed's notes Example 21.30 he discusses an example to use representing space to represent a non-representable sheaf. The goal is to show $\mathcal{F}(M) = \text{Map}(M,\mathbb{P}\mathcal{H})$ which shows $\mathcal{F}$ is representable by $\mathbb{P}\mathcal{H}$. He uses the phase "in essence $\mathcal{F}(M)$ is the space of smooth maps $M \to \mathbb{P}\mathcal{H}$". I don't quite get the reasoning behind his "in essence". Could somebody please help? For your convenience I reproduce the example below
