What does the line "edge resonance between an inductor and a capacitor" exactly mean?
All circuits have parasitic inductance and capacitance. Nothing can really be done about that. But, it is possible to find ways to use the parasitic components to advantage in some cases. In power conversion it is sometimes possible to arrange things such that the transfer of energy from a parasitic inductor to capacitor results in greatly reduced switching losses in transistors and diodes. 

The most simple example of edge resonance is a voltage step driven RLC with diode circuit. 



You probably know that the Q of this circuit is \$\frac{Z_o}{\text{R1}}\$, and as long as R1 has a value that allow Q > 5, the circuit will be very under damped (highly resonant). If a step input is applied, voltage on C1 will ring up to ~ 2Vstep. At that time current in L1 will be zero. D1 blocks any discharge of C1. That's a resonant rising edge. Replace the diode with a well timed switch and you can get a resonant falling edge as well. This may seem a trivial example, but all edge resonance is based on the same dynamic. 

H-bridges that have an inductive load and switch capacitance can use edge resonance for Zero Voltage Switching. Operational conditions can be narrow, restricting load and input range, for complete ZVS. Extra inductance or capacitance may have to be added to reach the proper condition for a given input voltage or load. 

Also, the same resonance can be used to reduce high frequency spectral components for reduced noise and interference.