To which question? Why they rotate around, or why they can all be played with success up into high elo?
The answer to the first one is just the nature of a game that has balance changes. It's not so much that one champion becomes strong so much as stronger ones get weaker as they get nerfed. Then some that were a lower tier now becomes a higher tier, and thus the constant rotation. The goal is to have perfect balance between all champions, but that goal cannot truly be achieved because one change to even one champion or one item has the potential to indirectly change many other champions. I can explain more if you'd like.
As for being able to play anything to success in higher elos, this is really dependent on circumstance. In Solo Queue, most everything has the potential to be good. Even "bad" champions such as Mordekaiser have win ratios fluctuating around 50% or so in Diamond. People use Elise, the "worst" champion at the moment, in high elo, and she has roughly a 42-45% win rate. Does this mean she's not as good a pick as other choices? It could be. But despite this, she's not a bad pick.
Team games are a different story, because with a coordinated team, composing a team and an overall strategy is much more important than in Solo Queue where he with the most kills wins. At this level, there are very many champions that truly are weaker, more disadvantageous picks than others. And these do rotate, but coordination and communication can make some champions' kits fundamentally weaker than others.
Fiora is a good example. At one point, her high elo Solo Queue win rate was peaking around nearly 60%, while her high elo team win rate was about 37%.
As for where you are right now, play who you want. Play who's the most fun. I can promise you, if you main any champion, and you learn them inside and out, you will outshine peopl using the "OP" champions who don't stick to a main. Some people float around and play OP champions, then when they're not popular anymore, move on to the next OP champions. Some people find 5-6 they love and play nothing but them. It's up to you which path you pick.
I main Swain. I know tons about his playstyle and what I can and can't do. And even if I get counter picked, my knowledge of Swain allows me to be calm and make way fewer errors than people who just pick up Viktor because he counters Swain or he's popular a certain month. It's all your choice, really.