Understanding a Single Transistor Flyback Transformer Driver?
You're nearly there. 

This is part of the problem.


  I'm aware that this isn't the case since transformers need changing currents to induce currents in their other coils. 


While this is not wrong, it's not as useful as saying that transformers need voltage to induce voltage in their other coils. Currents do flow as well, but their relationship is more complicated than for voltage.

The feedback coil is in series with the base. This has the polarity (##) to drive the base to a higher voltage when the primary current is growing, when the primary has a positive voltage across it, and lower when it's negative. Once it starts to turn off, the feedback makes sure it turns off hard. 

The resistor ratio makes sure there is enough voltage on the base to turn it on initially, when there's no voltage from the feedback winding.

The only other part of the puzzle is why the transistor should start to turn off anyway. There are two things that can do this, and the first to get there will trigger the end of the 'on' phase.

a) A transistor with base drive limited by R1 will only have a limited collector current it can support. Once the primary current has risen to this value, any further rise will pull the transistor out of saturation, and the collector voltage will rise, reducing the primary voltage. This will further reduce the base drive by transformer action to the base winding, and the transistor will quickly turn off.

b) The flyback core will magnetically saturate at a certain primary current. This causes the inductance to drop, which increases the rate of rise of primary current dramatically. It will now quickly exceed the transistor's limited collector current, whatever it was, and mechanism (a) will complete the switch-off.

(##) Thanks to Jonk for pointing out in comments that the polarity you might try to infer from that diagram is wrong. The absence of the explicit polarity indicating winding start dots should be a warning that this might be the case.