Why does the voltage across the speaker oscillate in this circuit?
Whatever AC voltage you get on the base of Q1 will cause the dc bias current of Q1 (set by the 330kohm resistor) to rise up and down in sync with that ac voltage. This modulation of dc bias current will be amplified by Q1 and results in a larger undulating current in and out of Q1's collector.

As this current has a path directly through Q2's base the same will happen with Q2. Q2's collector will amplify its base current undulations and force that current through the loudspeaker. This results in a voltage across the speaker but doesn't necessarily mean sustained oscillations. You have to examine the circuit to see if the voltage on the speaker reinforces the original ac voltage on Q1's base.

You do this by looking at the way the transistor current in the collector is affected by the base current. If Q1's base signal rises, collector current also rises and this forces more current through Q2's collector which increases the voltage across the speaker.

This is called positive feedback and results in oscillation.