As Zotjen suggests, I think it's more basic than that. 16 year-olds are considered to be young adults (and in many countries are treated as such under the law). Under 16 means a whole raft of reduced privileges/ greater protections, depending on how you read it. You're a child. And children don't get to compete in the Olympics (or get married, or buy smokes or booze, or take out mortages, etc).*
But it doesn't matter, really, what the reasons are for the rule. The rule is in place for everybody, and entering a team in -- let alone hosting -- the Olympics means you've agreed to abide by the rules as a nation. Letting underage competitors onto your team is a breach of the rules and ought to be dealt with in the same way as doping violations, etc. Whatever, this is going to be embarrassing, if true.
o
* EDIT: I stand corrected on this one (see posts below). The Olympics, clearly, is lousy with kids.