Quote by InvalidID
That doesn't really help me intuitively
to see where the angular speed comes from.
Also, since you solved the differential equation of motion for a particle with a spring, does that mean that the solution only applies to simple harmonic motion with springs? Is there simple harmonic motion without a spring?

You can't always expect to have the luxury of that, I'm afraid. If you have ever solved a second order diff equation, you will know that the result is in the form of a trig function. You just have to accept that.
It doesn't have to be a spring. That equation of motion applies to
many physical situations  where a 'restoring force' happens to be proportional to displacement from an equilibrium position. A simple pendulum approximates to this for small displacements from the vertical, too. Why not look it up?????
BTW, the Negative sign in the equation is because the direction of the restoring force is opposite to the direction in which the particle has been displaced (i.e. back where it came from). If you are not familiar with equations of motion then I am sure you will come across them before too long and things may get clearer.