How does VSEPR theory explain the formation of different bond angles in PCl₅?
There are only five Platonic solids: 


tetrahedron, 
octahedron, 
cube, 
dodecahedron, and 
isocahedron.


Therefore five atoms cannot be arranged in five equivalent positions, unless all five lie in a plane.
Trigonal bipyramidal is the minimum energy arrangement of five charges on a sphere. See Kevin Brown's MathPages Min-Energy Configurations of Electrons On A Sphere (Stable link via The Internet Archive).

Also there are not five equal energy states as demonstrated in the lecture notes by Marcel Schlaf and Kathryn Preuss from the University of Guelph:
Chapter 5.5 (continued 2) MO Theory ICl4, PCl5. (pdf) (Stable pdf link via The Internet Archive)