I do not know if I am allowed to answer your question by linking a video that goes into detail very thoroughly, and I apologize if I am not.
That said, this video answers your question perfectly, because it goes into detail the actual physics behind why the video of the rally looks the way it does. It has to do with how cameras work, especially the zoom feature. In a nutshell, the news cameras that were at the other side of the gymnasium were elevated and had to zoom in extremely far, in order to capture Hillary and have her take up most of the screen (requiring narrow lenses). This results in distortion for everything in the background because of the very narrow lenses used in the news cameras, compared to the very wide-angled lenses in the phones of the people in the video. That is why everything looks so small in the phones of the people in the crowd, as the news cameras were not using wide-angled lenses, being so far way and all. What you see in the phones of the crowd is what you would see if you were standing right beside them (not that you would see everything tiny, but that you would have the same perspective that their phones have; it is tiny because the cameras are zooming in from far away), if that makes sense. I think I have already went ahead and answered your question, even though I meant to just give you the video as a reference, haha, but oh well. Here is the video that goes into more detail: https://www.youtube.com/watch?v=3R-6wMrh8Ew.