I am trying to understand the basic principles of scalar diffraction theory and came across the sentence "E and H obey the Helmholtz equation", what does that mean? Does that mean that each component of E and H field obey the same equation?
Yes. First, $E$ and $H$ (each separately) obey the same wave equation which says nothing else that the $E$ wave travels with the speed of light and has the same wavelength as the $H$ wave which, too, travels with the speed of light. The Helmholtz equation is obtained from the wave equation when one separates space and time variables to find particular solutions.