Driving frequency is the frequency of the driving force. The driving force is an external force applied to the oscillator.
You typically only worry about "driven" oscillations when there is some sort of damping force--an undamped simple harmonic oscillator would simply increase its amplitude whenever a driving force is applied.
For example, let's say my oscillator is a damped oscillator, such as a mass resting on an incline (with some friction) attached to a spring (and just assume it's only allowed to travel up/down the incline radially to/from the equilibrium position with no angular motion relative to the equilibrium position). Without a driving force, the oscillator's response would be transient--i.e., no matter how much initial displacement you give the oscillator, friction will leech energy from the oscillator and it will eventually approach the equilibrium position. However, with different kinds of driving forces, you could make it approach equilibrium slower (or faster--here the driving force is just used as additional damping), you could have the amplitude of oscillations increase to infinity, or you could exactly balance the damping so that the oscillator maintains constant amplitude.
Just to elaborate, let's focus on that that last one--exactly balanced damping and driving. As the mass makes one complete cycle, a definite (and easily solved) amount of energy is lost from the oscillator to heat, via friction. Let's call the lost energy E.
But since the oscillator is maintaining a constant amplitude/frequency, we know that it is somehow getting that energy back. Since the only possible source of energy is the driving force, we know the driving force imparts exactly E to the oscillator every cycle.
When the imparted energy, Ei, is not equal to the energy lost to friction, Ef, the amplitude won't be constant. If Ef > Ei, the oscillator's amplitude approaches 0, and if Ef < Ei, the oscillator's amplitude approaches infinity.
It's possible for the imparted energy to come from any kind of driving force function such that the work integral equals Ei over a cycle. Among these are possible sinusoidal driving forces (which have well-defined "driving frequencies"). For example, a sinusoidal driving force with the same frequency as the oscillator could impart the desired Ei (this would have the textbook sinusoidal frequency) or an instantaneous jolt once per cycle could impart Ei (this would have a "frequency" equal to the oscillator--but the Fourier composition of this jolt would be many different textbook sinusoidal frequencies.) However, sinusoidal or other functions without the same frequency as the oscillator could also give the desired Ei-- an easy example is if the driving force gave TWO instantaneous jolts per cycle with (1/2)Ei each; this driving force would have a driving frequency equal to double the oscillator's frequency.
The function for the driving force, F(t) = Fcos(wt) is nothing more than the specification that the driving force actually be sinusoidal. They did not derive this (a driving force can be any old force function), they're just telling you to use a cosine-form driving force that has frequency w_d and amplitude F.
