Two posts that aren't weeks apart!? What is this madness? Eh, whatever, I don't have a lot of time to do this, so lets just get going, shall we?
Trope of the day:
Basically, the hero of your show or whatever is dismissed because of his age. Of course, the "kid" is almost always more competent than the person who's dismissing them. Complete unreality of this fact aside, it's fairly obvious why you would use this trope, especially in a show geared towards a younger audience: kids want to be acknowledge by adults. When you were seven years old, you probably wanted to do all the things that adults do; driving, staying up late, etc. Of course, your parents probably said you can't do that because you aren't a grownup. And in retrospect, they probably have a point. If someone thinks that they can be a robot when they grow up, you probably don't want them behind the wheel of a minivan.
While in real life children usually aren't more competent than adults (although some are), seeing them on tv allows them to live out their dreams vicariously. It's really more a form of wish fulfillment more than anything else. Children like seeing someone in their position actually getting the better of the "grown-ups" because they don't usually get to see it happen. It's a novelty, and allows them to imagine themselves in that position.
On a larger scale, there's the fact that a lot of people just like to root for the underdog.
From a story standpoint, someone who has to overcome adversity is a more interesting character than one who accomplishes their goals with no problems. A movie about a guy who wants to run in a race has a lot more punch if the guy has only one leg. (And even more punch if he also has a hook hand and a parrot, but that's beside the point).
At its core, this trope is about overcoming adversity, and age discrimination is as good a barrier as any.
P.S. This post doesn't have enough pictures. Therefore: