Sampling Theoreum: it is of course mathematically derived, but if you think of it from an intuitive perspective, suppose the highest frequency we want to sample is f_max, then of course we just need two piece of information in order to pin down the amplitute and phase of f_max. You don’t really need all those math stuff to understand it. You just need to pieces of information to pin it down, therefore, you need to samples for f_max.
Temporal Filter as Aperture: temporal filters can be seen as a aperture with varing shapes. For instance, and integrator is just a slot of 1, a differentiator have only two points, +1 and -1.