One intuitive way I've seen to think about the math is that if you are at any position inside the hollow spherical shell, you can imagine two cones whose tips are at your position, and which both lie along the same axis, widening in opposite direction. Imagine, too, that they both subtend the same solid angle, but the solid angle is chosen to be infinitesimal. Then you can consider the little chunks of matter where each cone intersects the shell, as in the diagram on this page:
You still need to do a bit of geometric math, but you can show that the area of each red bit is proportional to the square of the distance from you (the blue point) to it--and hence the mass of each bit is also proportional to the square of the distance, since we assume the shell has uniform density. But gravity obeys an inverse-square law, so each of those two bits should exert the same gravitational pull on you, but in opposite directions, meaning the two bits exert zero net force on you. And you can vary the axis along which the two cones are drawn so that every point on the surface of the shell ends up being part of a pair like this, which leads to the conclusion that the entire spherical shell exerts zero net force on you.