I believe that it is simply a more emphatic way to give the very same command. The politeness of this inflection seems to be dependent on the tone in which the command is given (perhaps rank/level plays a role too). For instance, your two examples seem pretty neutral, but I have also heard things like:
EDIT: My first example was erroneous and actually an example of the 出て行って contraction which translates as "get out and stay out!", and can apparently be further shortened to 出てって
待ってて - Wait! ~ "Hey guys... waiiit!" (spoken with more of a annoyed whine)
As such, this seems to be more of a spoken construct, allowing for tone to help convey the fully intended "politness."
EDIT: I think it does make the command come across more as a DEMAND; not doing what is asked in a 「〜てて」 command will likely lead to unfavorable outcomes (the kid with the toy would cry and pester the father, the person that requested you to shut up will get more frustrated as you continue to talk, etc).
I found this article which has an example comparing and contrasting two sentences, one using 見て and the other 見てて:
「見て！あの雲、ハートの形してる！」 Look! That cloud is shaped like heart!
「すぐ戻るから、テレビでも見てて。」 I’ll be right back, so watch TV or something.
But still, this doesn't quite fit with what I imagine/interpret that little girl is saying in your video... although perhaps it translates to something like
「見てて」 Hey, look at this/me/it for a minute, please...