Why is this game so dependent on the meta game.

The meta changes all the time. A Meta makes it possible for complete strangers to work together and achieve a goal. Without it, there'd be chaos. The meta is always changing - but it's not always right. The problem arises when people start saying "the meta is the only way to play/win." This is simply not true. But, following the meta is the best way to have a generally good experience on the competitive level of play. If people generally stick to the meta, people know how to play with their team. They know what to expect, and they know how the team dynamic will roughly play out.

Edit: to support my first statement, lots of elements in the meta have changed. For example, It used to be meta that only tanks/duelists took top lane. Now, we see durable mages head up there as well. In Season 4 and 3, Jungle would have practically been a troll pick. For a bit here in S5, she was the new top-tier jungler. AP Ezreal bounces on and off in the meta. Things like this are always evolving based on what changes Riot throws at us and what new strategies the professional players are doing.

Because at some point Riot started building their characters around the meta the playerbase created. When new champions started creating large deviations in the meta , nerfs or changes to push them back to intended roles, until people just basically stopped trying.

It happened as players figured out how to maximize the gold/experience gain. Same reason why the tallest player on a basketball team typically plays center or post and the best ball handler is usually the point guard. It provides a strategic framework for a group of strangers to turn into something that has a solid chance at winning.