Consider the Poincare map $P$, for which we place the Poincare section transverse to the trajectories and it intersects at various points which forms a set of discrete points.
For $\dot{x} = f(x)$ where $x$ is an $n$ dimensional vector, why the section $S$ is an $n-1$ dimensional surface? How can I visualize this?
Two angles for explanation:
The Poincaré map is supposed to remove exactly one dimension from the dynamics: The one corresponding to time. All other dimensions must still be present in the Poincaré map. Hence it’s state space must have dimension $n-1$. As the state space consists of points of the Poincaré section, the section also must have dimension $n-1$.
For most applications, you want the Poincaré map to have one iteration per oscillation¹. Hence, the Poincaré section has to be frequently intersected. Now, very loosely speaking, a trajectory segment (a one-dimensional object) has a probability of zero to hit any object (the Poincaré section) with dimension $n-2$ or lower, while it cannot miss an $n-1$-dimensional object. Of course, for some dynamics you may carefully place an $n-2$-dimensional section such that it is hit for sure, but this would mean that your dynamics has a spurious dimension.
¹ otherwise it’s every other oscillation, every high-amplitude oscillation, or similar, but the argument stays essentially the same