So long as your kid can handle him/herself for a few hours, what's the big deal? I believe parents are allowed to go through security to meet their kid at the gate, so really all the kid has to do is board, entertain him/herself, and deboard.
As to why, specifically? Well, suppose you split custody of a kid with your ex-spouse, and you live in L.A. while ex-spouse lives in New York. Or, suppose your parents want your kid to visit, but you live in L.A. while your parents live in New York, and you can't take time off from work (or you want your parents to get a little time to monopolize the kid).
When my parents split up in the early 70's, I lived in San Diego and frequently flew by myself to Seattle (from age 8 til adult). It was a great experience, and I wouldn't trade it for much. Now as a father of two daughters, I'd be a little less cavalier, but I'd still want them to stretch their wings and learn a little resilience/independence.
As to why, specifically? Well, suppose you split custody of a kid with your ex-spouse, and you live in L.A. while ex-spouse lives in New York. Or, suppose your parents want your kid to visit, but you live in L.A. while your parents live in New York, and you can't take time off from work (or you want your parents to get a little time to monopolize the kid).