The best way to see this is to draw the transition diagram for the Markov chain. It's best to do it so that the states 1,2,3,4,5 appear in a circle. This type of diagram is given in the solution to this question.
Looking at the diagram, you can see that if you rubbed out state 1 and changed it to 2, changed 2 to 3, 3 to 4, 4 to 5 and 5 to 1, you'd still have the same diagram as you had before. You can keep doing this (switching each state in the diagram for the next higher one) and the diagram keeps on staying the same.
This is the symmetry of the situation and it means that each state is equivalent in the long-run. If the states are all equivalent in this way, we expect them to have the same long-run probability giving the solution (0.2, 0.2, 0.2, 0.2, 0.2).
The alternative is to solve the matrix equation pi = pi*P as we ordinarily do to find stationary distributions. This should get you to the same place, but would probably take longer than spotting the symmetry.
