The angular frequency is a rather abstract concept which deserves some discussion.
At the most basic level, from , you should appreciate that the angular frequency ω is simply the frequency of the oscillation multiplied by .
At a more abstract level, from , we can think of as the phase angle of the oscillation. So ω is the “velocity” at which the phase of the oscillation progresses.
The angular frequency ω in SHM is actually very similar in concept to the angular velocity ω in circular motion. (This explains why they are both given the symbol ω.)
For oscillations, is the rate of change of phase angle.
For circular motion, is the rate of change of angular displacement.
Lastly, a circular motion collapsed into one dimension is actually an SHM. For example, the displacement in the y-direction of the circular motion illustrated below is actually an SHM. The circular motion’s radius R, is the SHM’s amplitude R. The angular velocity ω of the circular motion is also the angular frequency ω of the SHM.
Circular Motion vs Oscillation