The advice comes from two good things and one bad thing:
1. We take self-incrimination more seriously than most countries.
2. Hearsay is inadmissible in most circumstances. (I think this one's true in most of the world.)
3. Our police are fucking lunatics.
In most of the world, you might have trouble relying on a fact that you didn't give to the police right off the bat, but here, that can't be used against you. What's more, any such fact can't be used to help your case, since that would be hearsay. So if you're innocent, it would be at best harmless to talk to police, but that's where (3) comes into play. Remember, one in every hundred Americans are in prison right this second, and that's not counting those who have been released, and will soon be, if not back, illicitly employed in one way or another because they'll never have a "real job" again. What's more, the "Reid technique" (basically, making you feel guiltier about letting the police down than committing the crime) is legal in the US, which has been shown to not only get false confessions, but convince people who could not possibly have committed a particular crime that they have.
The police will not help you, unless your goal is to have someone else arrested and/or shot. Wait until you're talking to the DA.