Let be $${\bf F}=-y\hat{\bf i}+x\hat{\bf j}$$
In this video how did the professor figure out above field has angular velocity of $1$?
I understand $(-y,x)$ rotates the entire plane counter clockwise by $90$ degrees. So this vector field seems to rotate in counter clock wise direction at uniform speed. But how to work the actual angular velocity?
The vector field corresponds to the system of differential equations $$ dx/dt = -y ,\qquad dy/dt = x , $$ which has the general solution $$ x(t) = A \cos(t-\varphi_0), \qquad y(t) = A \sin(t-\varphi_0) . $$ From these formulas it follows that every solution curve (except the equilibrium point at the origin itself) is a circle centered at the origin, and that the period to go around such a circle is $2\pi$ (regardless of the radius).
You can also see this by expressing the differential equations in polar coordinates $x=r \cos \varphi$, $y=r \sin \varphi$, which after some computation gives $$ dr/dt = 0 ,\qquad d\varphi/dt = 1 . $$