How does a flute make different sounds?

A flute makes different notes by covering different holes along its body, which changes how much of the air column inside the tube is vibrating. A shorter vibrating column of air makes a higher note, and a longer one makes a lower note, so covering more holes near the far end changes the pitch.

Warming the flute up doesn't change its notes in any noticeable way, and shaking it would just make noise, not music, so neither of those explains how a flute actually works.

Many wind instruments, like clarinets and recorders, use this exact same trick of covering holes to change the length of vibrating air, which is why they can all play a full musical scale.