We are our experiences
People are often shaped by their experiences, our views and personality are carved by our past. A character in an RPG is the same, their outlook on the world might shift depending on what happened to them in the past. In a setting with an existing lore, weaving your background into it can help tell you how your character might feel or react to a given situation. For example in Warcraft, a human character's opinion on the Scourge and Orcs might be different depending on whether they grew up in Gilnaes, Stormwind, or Lordaeron. I find that often my background tells me more about how my character might react than my initial concept does; it can provide goals and opinions that I might not have considered. Plus it's a lot of fun to try and figure out how your character weaves into the overall story.
But you are the things you do, not the things you've done
Some players fill their background with things that might make their character seem amazing. They're the bastard child of Arthas and Jaina, the last hatchling of the Pink dragonflight, or Tirion Fordring's former squire. These are all cool things to have done, but do they really help your character come to life? I think that a well played commoner is much more interesting than a poorly played king. As a player I don't much care if you are the son of a baker or the Prince of Kul Tiras, I care about how you play. Not to mention that some of those more "potent" background stories often tend to be compensating for an otherwise lackluster character, are an attempt to make a Sue-ish character, or are an attempt to shoe-horn themselves into plots. Focus less on making yourself an amazing special snowflake and let your character speak for him or herself.
It's for OTHER people too!
But really, the background is just as much for other people; I like to think of it as a series of hooks that other people can latch on to. So many players write backgrounds with a sense of "Hands OFF" t them, and certainly you are allowed to decide what other people can involve themselves in, but the background can be a springboard for other players to create connections and roleplaying opportunities. Maybe you and someone else both were at the Battle of Icecrown...suddenly you have a connection and maybe you can play that out. Maybe you both studied at Dalaran in the "good old days" and can reminisce about how beautiful Hillsbrad once was. It can be scary to put your character out there, but it can also be an amazing way to create stories you might have never thought of.
Nothing in here is exhaustive, only some suggestions; I have seen players with VERY "special snowflake" backgrounds play their character very well. I recall an extra-dimensional deity in Stormwind once, but the player handled it well enough, even if I did think him odd at the time. The background can be a handy tool, and it can be a lot of fun to collaborate with others to weave them together...worst that can happen is you choose not to.