How does this 4-transistor DC-DC 1.5V to 9V boost converter work?
Q3/Q4 in conjunction with the voltage divider R4/R5 and LED are supposed to regulate the output. Q4 should switch with about 0.55-0.59V on the base, so around 10-12V on the output by my estimation, assuming the LED drops 2V at ~600uA. 

The oscillation is supposed to be by Q1/Q2 and C1, with off time controlled by \$R1 \cdot C1 \approx~ 10us\$. On time in the original design could be limited by inductor saturation more than anything else- the current increases linearly until the inductor saturates, the current spikes up and Q2 comes out of saturation, causing the pair to more-or-less snap off, or by the current supplied by C1 tailing off, causing Q1 collector current/Q2 base current to to drop and again Q2 comes out of saturation, or some combination of the two. 

Edit: These self-oscillating transistor converters tend to be rather disappointing in performance and efficiency. 

Consider using a nice well-designed boost converter chip such as an AP3015A (2 cells) or LM2623 (will work with 1 or 2 cells).