What causes electromagnetic waves to propagate in free space? In free space, $\rho=0$ and $J=0$, so there are no electromagnetic sources/sinks. Maxwell's equations thus reduce to:
$\nabla\cdot E = 0$    
$\nabla\cdot B = 0$
$\nabla\times E = -\frac{\partial B}{\partial t}$
$\nabla\times B = \mu_0\epsilon_0 \frac{\partial E}{\partial t} $
Suppose I was writing a simple simulation to visualize the electromagnetic field in free space. I have seen people talk about waves propagating in free space, and I know that there is no such thing as electromagnetic waves being created from nothing -- it is usually assumed that such electromagnetic waves propagating in free space are plane waves that originated in a charged source extremely far away. 
However, when I actually implement the "oscillation" in the EM field, where does that oscillation come from -- speaking practically from a coding point of view? Do you just hardwire, e.g., a sinusoidal source at the location you are interested in probing the EM field? 
And if there were no such "magical" waves propagating in free space, would the EM field just remain smooth, without any oscillations, vibrations, sinusoids, etc.? In other words, can you have a completely stationary electromagnetic field, or would the last two of Maxwell's equations above prevent such stationary EM fields? But then what, in free space, would cause the initial change in the electric or magnetic field to get the oscillations going?
To put it one last way: suppose I wrote a simulation involving the 4 Maxwell's equations above (free space). Would the EM field be stationary for all time, and the only way a propagating wave would appear is if I perturbed, say, the electric field which activated a never-ending loop of the curl equations? So if the initial values of E and B were both 0 in my simulation, then they would stay 0 for all time. But if one or both initial values of E and B were non-zero, then the curl equations would be "activated" and result in a never-ending loop of oscillations? 
 A: I think the answer is simply: "Yes".
What you should keep in mind is energy conservation: As long as there are no sources, the total energy of the electromagnetic field is conserved.

But then what, in free space, would cause the initial change in the electric or magnetic field to get the oscillations going?

A source, which is possibly localized somewhere and not necessarily non-zero at all times.
A: The question "What causes electromagnetic waves to propagate in free space?" is essentially the same as asking why do electromagnetic waves exist. For that matter, why do electrons exist or why does anything exist at all? Rather than digressing into philosophy, let us focus on physics. We should remind ourselves about how profound the concept of a field is. This concept originates from Michael Faraday. A field assigns a number or tensor to each point in space. In other words, a field is a property of space itself. When a field has the lowest possible value, we call it a vacuum. Empty space is not nothingness. "Empty" space has structure. In fact, very rich structure. In the case of electromagnetic fields, this structure is described by Maxwell's equations. An electromagnetic wave is literally a property of space (with time dependence). Let us suppose that the fields of electromagnetism are disturbed. A disturbance could, for instance, be caused by an electron colliding with an atom. The fields (more properly, a single field called $A_\mu$) has a precise response to this event. A disturbance of the field propagates outward at the speed of light from the location of the event. The disturbance is described by Maxwell's equations. In other words, space itself has a structure that we perceive as an electromagnetic wave. Once the electromagnetic wave is created, there is no need for sources.
This is more or less a simplified viewpoint of modern physics. All fundamental particles are quantized excitations of different fields. The electron is a quantized excitation of the electron field. A quark is a quantized excitation of a quark field and so forth for all the other particles. Different fields correspond to different properties of space. All matter, energy and fundamental forces are quite literally properties of space itself. All of physical existence is derived from space. Fields are indeed profound!
