How do you edit out pauses in YouTube videos?
The effect you are talking about is called jumpcut and it is particularly popular among vloggers.
As far as I know, there is no automatic way to do it although it would probably be possible to do from technological perspective.
However using jumpcuts has several functions.

You get rid of silence and squeeze as much information in short form as possible, keeping viewers attention.
Second important function is selection of your shots. While going through your material (shot usually in one take) you can discard stuff you don't want (even if you were talking at that point) and fundamentally alter the tone of your video.

This increased creative control is worth it considering that editing your video manually in this manner is very easy and won't take much time or effort.



tl;dr
It might be doable, but there is no user friendly solution now. Anyway I would suggest to do it manually for various reasons described in full answer.