How does the 'phase' of a cosine wave become 'pi' when it transitions from positive to negative?
The statement in the video is, at best, misleading. Normally, when people speak of the "phase" of a sinusoidal function $\cos \omega$, they are talking about the value of $\omega$. For instance, when we say that a wave undergoes a "phase shift", we mean that the value of $\omega$ has changed. More information on this definition of the word "phase" is at https://en.wikipedia.org/wiki/Phase_(waves).

As noted by Jazzmaniac in comments, we could also try to interpret the statement by regarding the value of $\cos \omega$ as a complex number $z$. In polar representation we can write $z = |z| e^{i \phi}$ where $\phi$ is the phase of the number $z$. This use of the word "phase" is misleading in this context. For instance, applying a "phase shift" would mean that we multiply $\cos \omega$ by some complex number $e^{i \phi'}$. That certainly wouldn't implement a phase shift of the sinusoidal wave, though, in the usual sense.