It's actually a little tricky without some advanced ideas. The main idea is this: a polyhedron should be bounded, meaning it cannot stretch off an infinite distance in some direction. Since it is bounded, it is contained in some sphere, for a sphere large enough to contain it. Now it should be evident that a sphere can be chosen so that it intersects every corner of a regular polyhedron. As an aside, can you figure out how many points in the surface of a sphere in 3-D space will uniquely determine the sphere? That is, what number n is such that if n points are carefully chosen in 3 space, there is exactly one sphere that goes through the points, but for many choices of n+1 points there is no sphere that goes through them?