https://en.wikipedia.org/wiki/Buck%E2%80%93boost_converterIt is a little hard to answer this question since power converters are a broad topic. We did a whole subject on them at Uni in fourth year. We did SPICE simulations, built circuits, measured waveforms etc.
I can try to provide some intuition though. Think of an inductor as having a high voltage across it when the current through it suddenly changes (v(t) = Ldi/dt). If the switch is in the off state then the inductor current "sees" the load through the action of the diode. If the duty cycle is say, 80%, the inductor current will "see" the load for 20% of the time, or a short amount of time. The "spikier" this current spiking is, the higher the voltage.
I hope that helps. I can see why you might be confused. Since everything appears to be in parallel with the source, it may be tempting to think that everything sees the source voltage. However, notice that the diode forces the inductor to conduct during the on state, while not passing current to the load. In the off state, the inductor's voltage actually adds to the load, due to the reversed polarity of the output, due to the diode.