Why does changing the sampling or playback rate of a sound changes its pitch?
Sample rate or playback rate is just the speed at which your software or hardware reads through the audio buffer and plays back what it finds. Just adjusting the sample rate speeds up or slows down the rate at which the waveform is read, which effectively shortens or lengthens the waveform contained within.
Consider a 440 Hz Sine wave. If you play that sine wave at half speed, you are playing a 220 Hz sine wave, which results in a tone an octave lower than your original sound.