They have to do with the behaviour of the electromagnetic field inside of different materials: https://en.wikipedia.org/wiki/Electric_displacement_field , https://en.wikipedia.org/wiki/Magnetic_field#The_H-field

To get Maxwell's equations in a vacuum, just replace D with εₒE and replace H with B/μₒ.

Apologies - I didn't even realize that the epigraph expressed some of the equations in terms of flux density! That's what those extra terms mean! I did actually include an explanation of what flux is and how to intuitively derive it within the guide, and you can find it near the integral explanations. Hopefully that helps!