Electromagnetic wave from Maxwells equations in free space

[Melac12]

My textbooks says in a region where there is no charge or current Maxwell's equations read
divergence of E=0
Curl of E=-dB/dt all d are partial
Divergence of B=0
Curl of B=ue(dE/dt)

I get the math of showing that there are waves, but I don't get some of these conditions. 1st don't you need a charge in the 1st place to create an electric field? and then doesn't that charge have to move to created a magnetic field, which would be current? I mean with no charge and no current what makes the electric and magnetic field?
I hear that light can be made by taking an electron and shaking it. Electron being an electron has charge and shaking it make the magnetic field. If that not right one demonstrate light form electron?

 

[M Quack]

You do need charges to create an electromagnetic wave. However, even if you are in a perfect vacuum far away from that charge you can still have electromagnetic fields which will satisfy the conditions you cite above.

 

[Born2bwire]

The differential equations operate locally at a given point in space. While you need a charge or current to generate the electromagnetic waves, you can observe them in regions where there are no sources. The integral form of Maxwell's equations represent integrations over surfaces and volumes and make it easier to see how the equations operate in regions containing sources.