As pointed out by Sam Hopkins in a comment above, secondary polytopes can be seen as a particular case of the fiber polytopes of Billera and Sturmfels (https://doi.org/10.2307/2946575). 
This fiber polytope view provides the answer (or, at least, one answer) to what *each point* in the secondary polytope represents: 

A point in the secondary polytope of a point configuration $A=\{a_1,\dots,a_n\}$ corresponds (not uniquely) to a **section** of the natural projection
\begin{align}
\phi: \Delta^{n-1} &\to \operatorname{conv(A)}\\
e_i &\mapsto a_i,
\end{align}
where $e_1,\dots,e_n$ are the standard basis in $\mathbb R^n$ and $\Delta^{n-1}$ is their convex hull, that is, the standard $n-1$-simplex. 
More precisely, the projection map $\phi$ induces a (non-surjective) map from the set of all sections to $\operatorname{conv(A)}$, sending each section to its average, and the secondary polytope is the image of this map (perhaps scaled, depending on your choice of normalisation).

Vertices of the secondary polytope are special in that each of them is the image of a unique, and extremal, section, obtained by ``coherently'' picking the vertex of each fiber in a fixed direction. The section obtained in this extremal case is a union of faces of $\Delta^{n-1}$ forming (via $\phi$) the regular triangulation of $A$ corresponding to that vertex. This is why regular triangulations are also sometimes called *coherent*.

Note: I am writing things in the language of point configurations. For a polytope $P$ as in the original post, the point configuration $A$ is the vertex set of $P$.