What is the significance of angular frequency $\omega$ with regards to wave functions? What is the physical significance of $\omega$ in a function like
$$
f(x) = A\sin(kx + \omega t).
$$
The only place that I am familiar with angular frequency is when dealing with circular motion, but when it comes to waves I am not clear how the analogy holds. I am familiar with the idea of a reference circle when dealing with simple harmonic motion, I am still a bit unclear on how this extends to waves. What exactly is this quantity measuring, and why is it useful to write the function in this form rather than an alternative (but equivalent) fashion?
 A: One of the most useful ways of describing SHM is obtained by associating it with the projection of uniform circular motion.
Imagine that a disk of radius $\mathit{A}$ rotates about a vertical axis at the rate of $\omega~\text{rad/s}$. Also imagine that a peg $\mathtt{P}$ has been attached to the edge of the disk and that a horizontal beam of parallel light casts a shadow of the peg on a vertical screen. Then this shadow performs simple harmonic motion with period $\dfrac{2\pi}{\omega}$ and amplitude $A$ along a horizontal line on the screen.

By geometrical projection it simply means the process of drawing a perpendicular to a given line from the instantaneous postition of $\mathtt{P}$.

Now, let the center of the disk be $\text{O}$. Then the end point of the rotating vector $\text{OP}$ can be projected onto a diamater of the circle. Suppose we choose the horizontal axis $\text{OX}$ as the line along which the actual oscillation takes place. The instantaneous position of $\mathtt{P}$ is then defined by the constant length of radius $\mathit{A}$ & the variable angle $\theta$ that $\text{OP}$ makes instantaneously with $\text{OX}$. Using the convention for polar coordinates, if we take the counterclockwise direction as positive; the actual value of $\theta$ can be written:$$\theta = \omega \cdot t + \alpha$$ where $\alpha$ is the value of $\theta$ at $t = 0$. So, the displacement $x$ of the projection of $\text{OP}$ along $\text{OX}$ is given by:
$$x = \mathit{A} \cos\theta = \mathit{A} \cos({\omega\cdot t + \alpha})$$.

Courtsey: The M.I.T introductory physics: Vibrations & Waves by A.P.French

How is it related to wave?
Suppose in the wave motion, $\psi$ is the quantity that varies with position and time giving rise to wave. Thus $\psi$ is the wave function. Since, it depends on space and time, we can write it as $\psi(x,y,z,t)$. For convenience, just think about one-dimension. So, then the wavefunction is $\psi(x,t)$. Now, its variation does invoke the wave. Wave can be of any form. But the most fundamental wave is pure sinusoidal wave. So, the attention will be overwhelmingly to this sort of wave; this does not make any discrimination as any complex disturbance that repeats itself regularly with a certain period can be built up from a set of pure sinusoidal vibrations with appropriately chosen amplitudes. 
Every sinusoidal vibration arises due to restoring force that are proportional to the change of $\psi$ from equilibrium. Thus $\psi$ varies sinusoidally which means it is undergoing SHM. Now, since $x$ is any arbitrary point, every point in the wave is undergoing SHM. Now, since SHM can be associated to the geometrical projection of uniform circular motion, we can write:
$$\psi(x,t) = \mathit{A}\cos{(\omega \cdot t + \alpha)}$$. That is how, the wavefunction is connected to $\omega$.

You can see in the picture how the bead is moving sinusoidally. This is, in fact, SHM which can be associated with the geometrical projection of uniform circulation whose angular velocity is $\omega$.

Physical significance of angular velocity:
$\omega$ is the angular velocity of the circular motion which is associated to the SHM of the quantity that forms wave. 
Actually, $\omega^2$ is more meaningful. That is the square of the angular frequency of oscillation $\omega$ is equal to the return force/restoring force per unit displacement per unit $\text{mass}^1$.

$^1$ Mass does not always mean inertial mass as in the case of the electrical examples like LC circuit.
A: You could consider omega to be a pure indicator of periodicity in the cycle.  Larger omega gives you more rads per second.  Larger omega gives you shorter wavelength, and therefore more energy required to keep the cycle going.  You can look at omega in the equation and get an idea of the amount of energy you are dealing with.
