For some stories (and movie franchises) the reason to start off with an origin story is that of explaining the character to the audience without an infodump. How did that character acquire superpowers, money, fame, etc.? What made him or her into the person they were?
The definition of "origin story" on Wikipedia mentions that origin stories are retold again and again in order to "keep the characters current". The 1950 Superman is very different from the 2013 one. Ditto the radio version of the Lone Ranger from what will be appearing in theatres soon. But with the recent retelling of stories not even a decade apart I'm more inclined to think that it's because each writer and director wants to put their own spin on the character, their own agenda.
Robin Rosenburg says an origin story shows viewers how to become heroes or how to cope with adversity. TV Tropes goes into a bigger list than the previous article (which had three) of the origin types of superheroes and their motivations. There is indeed a difference between "The Chosen One" versus "Hero by Accident" versus someone who purposefully sets out to help others. Each type of hero has their followers, as longtime comics readers can attest.
Not all bigger than life characters need an origin story. Some develop one along the way. Viewers went through two incarnations of the Doctor before learning how he left Gallifrey and even now, during the 50th anniversary of Doctor Who, the complete origin story is still murky.
When the original television series of Star Trek began, viewers saw the regular shipboard adventures of "every person" crewmembers. However, with the 2009 movie makeover, an origin story appeared. Captain Kirk became "The Chosen One" rather than someone who had trained through Star Fleet Academy and risen through the ranks like everyone else.
With YA stories, you are often starting with a character at the time they are deciding to become someone, though the Harry Potter series did start with baby Harry being dropped off at his relatives. There is a reason many YA fantasy stories are labeled "Coming of Age" tales.
If you're just telling a regular story, one theory is that you're supposed to start when "things change". That way both the character and the reader can try together to make sense of what is happening in the character's life. It's up to the author, then, to decide when the story starts. How far back do you go in the character's life? With a superhero, it can be when she or he first acquired their powers, or first decided to fight evil. But with a regular person, do you begin when they first decided to change things? Or when change happened? Do you begin when he or she began school? When they were born?
It’s hard to resist wanting to explore important decisions in your characters' lives. When I wrote The Crystal Throne, my point of view characters met all these interesting people. But those characters' back stories were not important to the story. They were simply there, at that point in their lives. It was only later in short stories that I began to explore why a Fleet One would be interested in legends, or what happened in a young girl's life that would lead her to become a wizard, or even where a particular ring came from.
What origin stories interest you? When do you say "Enough about the origin? What else is happening with that character?"