I can't say who exactly is responsible for 'you got this'. It just began to grate at me the year when every new movie or tv show had a character saying it.

A similar phrase is 'I need you to', which appeared around the same time. Eg: 'Okay, I need you to calm down' or 'I really need you to be supportive right now', etc. To my ears it sounds gratingly self-entitled.