How to design wireless charging coils so they are resonant?
how do I design the coils so that they are resonant? It would be great
  to get an answer that would be easy to understand for someone without
  education in electrical engineering, if that's possible, as much as it
  is possible.


What you have without resonance is something like this: -



Image source

With a capacitor that resonates the receive coil you get a circuit like this: -



And this magnifies the induced voltage in the receive coil (\$V_{IN}\$ below) because it acts as a very resonant low pass filter like this: -



\$R\$ represents the coil and transmit driver losses.



\$V_{OUT}\$ can be made to peak at magnitudes like this: -



Interactive filter tool

With the values chosen, the circuit resonates at about 411 kHz with an amplitude gain (\$G_P\$) of 77.5 or nearly 38 dB. But, bear in mind, that when loading the circuit and taking real power from it, the peaking may drop to a few dB. However, even 6 dB (a doubling of voltage) is a great benefit for the selection of one or two parallel capacitors to hit the sweet-spot.

Also remember to use high-speed diodes in the receiver power rectifier. Something like the ubiquitous 1N400x type diode is wholly unsuitable because of its reverse recovery time.

You can also do a similar trick for the transmit coil but now, what happens is that you "tickle" a parallel coil and capacitor with a low current and get a much higher current. Sure, it can take several milliseconds to build up but who cares about that?