Negative frequency of a sinusoidal wave on the magnitude spectrum: Physical explanation? link to the video: Negative frequency

Must a cosine function be expressed as two exp functions with opposite frequencies/directions of rotation and half the magnitude/amplitude(1/2),
therefore resulting in two magnitude spectrum components?

Why can't there be one single component at w? 

I know this is very basic but there are only videos about the "how" not the "why".Thank you in advance!
Regarding your first comment, I have a figure that might help, in terms of intuition. It's a bit badly scaled but it might do the work.