At your dish, there's a feedhorn.
That feedhorn converts the focused waves (which are circularly polarized) from the parabolic reflector ("dish") to a wave in a (typically rectangular) waveguide with linear polarization.
That waveguide is pretty short and ends on the PCB of your LNB, where there is a waveguide feed, which is essentially a purpose-built antenna for the inside of a waveguide. From there, the energy is still in a waveguide, but that waveguide is a stripline or similar on the LNB PCB. That signal however, is measurable as current (or voltage) at the pads of components attached to the stripline – for example, amplifiers and mixers. They produce an RF current on their output pins, which directly is on a stripline again!
For RF, whenever the distance your signal has to travel aren't really really small compared to the wavelength (and satellite TV wavelengths are already pretty small), you need to do a waveguide to transport the signal. Everything else would just radiate or convert the signal to heat!
The LNB mixes down the signal to an intermediate frequency (still RF!) and outputs it on the center conductor of yet another waveguide – the coax cable running down to your TV / set-top box. At the receiving end, another coax connector "converts" the signal from the coax kind of wave guide to a PCB stripline, where it's again measurable as current for chips inside the receiver.
So, there's not really much of a "conversion" that happens between EM fields and currents on conductors: one only come with the other, there's no physical way around that. See Maxwell's equations for the physical explanation of that.
If you're an EE student, rest assured that you will have plenty of contact with Maxwell's equations soon. I had at least three lectures that focused on them in the mandatory part of my EE B.Sc., and that's the part all EEs have to take; I went on to do communications, so I had some more :)