Touga's explanation in the manga
In the third volume of the manga, before their second duel together, Utena asks Touga:
Why do you want it so much? "The power to revolutionize the world" … I've never understood it.
Touga's response to this is:
Oh. That's easy. Tenjou, have you ever felt that you were an outsider? That this place we look down upon … It's like another world, far away from the real one? Our true world exists someplace else. We all want to get that world back so we can finally return to our deepest selves. This world we inhabit — it's like an egg … A small world encased in a hard shell. The chick will die, unborn, if it doesn't break through that shell … Die without ever knowing that it's a bird! If we don't break the world's shell, we'll die without beyond born, without knowing we can fly! And I want to fly. That is why I will revolutionize the world … So that I can be myself!
(Italics are from original text. Some of the punctuation is my own.) This implies that in the Utena manga, this "power to revolutionize the world" is not necessarily about physical power. Rather, it is about being able to make a proper place for oneself in the world.
Can a similar interpretation apply to the anime?
The anime diverges from the manga somewhat in plot, but this interpretation helps clarify the anime's own discussion of the "revolution." We often are shown a sequence where the student council members say:
If it cannot break its egg's shell, a chick will die without being born. We are the chick. The world is our egg. If we don't crack the world's shell, we will die without being born. Smash the world's shell! For the revolution of the world!
This is very similar to the second half of what Touga says in the manga. There is the same notion of self-realization and overcoming the world here. Furthermore, Mikage's response in the anime after students recount their troubles — "Your only choice is to revolutionize the world." — shares the element of "revolutionizing the world" for personal reasons.
When Nanami substitutes for Touga, she introduces a variant in the sequence:
If it cannot break its egg's shell, a chick will die without being born. The chamber of freedom and the cage of freedom. Without revealing the vastness of the sky, both care for the chick. Smash the world's cage! For the revolution of the world!
The meaning of this is less immediately clear, but it does not seem inconsistent with the interpretations earlier.