Using the fact that every vertex 'u' is connected to d(u) (degree of 'u') faces of the polyhedron, we try and see what happens to |V|,|F| and |E| when a vertex is removed and a new polyhedron is formed with |V'| , |F'| and |E'| (# of vertices, faces and edges).

|V'|= |V|-1 (trivial)

|F'|=|F|-d(u)+1
all the d(u) faces vertex 'u' is connected to will merge into a single face

|E'|=|E|-d(u) (trivial)

observe that |V|+|F|-|E|=|V'|+|F'|-|E'|

repeat this process on the given polyhedron until only a cycle is left.
And we can see that |V|+|F|-|E|=2
(the face formed by merging the faces to be removed and the face already present is the same, i.e our cycle has two faces)