So, why does a Meta-game exist? Not just in League but in dozens of other competitive games Meta-games exist. I’m going to give my best insights into why the Meta-game exists, not just in League but across various e-sports and competitive games. I will also touch on Counter-picking and trying to find new 'Meta' champions.
First, let’s establish why characters/champions/build-orders/strategies become popular. We’re going to do that by approaching the game from Blind Pick/First Pick state as the effect is similar only with different goals. When Blind Picking into an unknown match up what champions are players most likely to lean on and why? Well, there are a few examples in most lanes. This is not intended to be an exhaustive list, merely demonstrative:

Mid: Gragas or Orianna
Top: Renekton
Jungle: Vi, Lee or Elise
Marksman: Caitlyn
Support: Thresh

In most circumstances these champions have favorable match-ups or at least few match-ups they lose outright. Often they possess elements that allow them to excel in their lanes; sustain, tankiness, wave-clear, CC, gap-closers and so on. They tend to be strong generalist picks that teams can either be built around or do not force a team into certain compositions. Due to this flexibility and strength these champions are commonly picked; this is in essence the birth place of Meta.

Gee Tawny, we already knew that. Maybe you did, maybe you didn’t. The key to a champion becoming popular in the Meta isn’t a professional player using them; rather it’s simply Flexibility. If a champion can do many things well with few bad match-ups they might well become a Meta pick.

Second, let’s look at and understand Counter-picking. Counter-picking is dangerous in Blind-Pick but a legitimate strategy in Draft after first pick. Counter-picking relies on choosing a character with a strength commonly weighted to one area; Counter-picks may be early-lane counters or late-game solutions to problem champion. Counter-picks may indeed be legitimate choices in their own regard though, don’t pigeonhole champions by thinking they are only a counter. Let me try some examples:

Enemy Top Picks: Mundo
Ally Top Picks: Trundle
Enemy Top Picks: Renekton
Ally Top Picks: Ryze
Enemy Support Picks: Thresh
Ally Marksmen Picks: Sivir
Enemy Marksmen Picks: Vayne
Ally Marksmen Picks: Caitlyn

Each of these examples one can argue is picked for a different reason which leads us to another pair of important points for counter-picking:

Know how you Counter, Know why you Counter.
If you don’t, then why are you counter-picking in the first place? The point of Counter-picking is to abuse the weaknesses another champion has. Because of this fact Counter-picking can be inherently more risky; especially if you are less skilled with your erstwhile ‘Counter’. Counter-picking is not a game of rock-paper-scissors, don’t treat it like one.
So why does Counter-picking happen at all if it can be risky? By exploiting an enemy’s weakness mercilessly or by negating one of their strengths you hope to create an advantage. Suddenly Thresh is a lot less scary in lane when his hooks are negated on a spell-shield. Vayne needs time to ramp-up to her late-game phase and if Caitlyn keeps leveling headshot after headshot it’s going to slow that ramp. By the same measure, if Caitlyn never harasses Vayne or Sivir get caught by every single hook then you’ve squandered your advantage by picking those champions for those purposes. Therefore Counter-picking is almost always more useful the higher the skill level involved. Without the mechanical and match-up understanding to abuse those advantages as well as make the most of the champion on its own it’s not a good option. If you intended to counter-pick make sure you understand the ins-and-outs.

Finally, running against the grain; so say you don’t want to pick a Meta-champion, maybe you want to break the Meta and be a trendsetter, how do you do it? Would you believe me I told you it was actually pretty easy?
It’s both easy and hard actually; it takes an objective look at what a champion can do and a deep understanding of the game, then it requires practice and patience to actually test your ideas with said champion. Let me give you a list off the top of my head:

What is the purpose of this pick?
What role do I want to fill with this pick?
Does this pick’s kit offer anything unique?
What are the strong match-ups for this pick?
What are the weak match-ups for this pick?
Is the performance of this pick on par or greater than with other picks?
If this pick is not on par or greater than other picks, are there unique circumstances where it excels?

If you can answer those questions it will likely help you determine whether a champion is 'viable' in the Meta-eviron of League and the more you do it the deeper you're depth of game knowledge will grow. That's not to say you shouldn't play what you like, Lord knows I do despite the consequences.

There a ton more things I could go into, Rotational Meta, Team Composition, and probably a dozen examples of other games but I really won’t bore you with them, this is becoming a bit Block O’ Text as is. I hope you found this informative, if you didn’t then I hope you at least will leave your thoughts in a comment below.

Sincerely, TawnyOne