Here's my best guess:
Start with the switching mode of a transistor; if you bring the base voltage close to the emitter's voltage the transistor goes to cutoff; only a little leakage current will flow. Bring the base voltage closer to the collector's voltage, and the transistor goes into saturation, and the transistor will only drop a small fraction of a volt and pass a lot of current.
In the satucure circuit, the primary of the transformer is connected to the collector of the transformer and the secondary is connected to the base.
When the transistor is conducting, current will flow thru the primary. As long as the field in the inductor is expanding, a current will be generated in the secondary which pushes the base voltage closer to the collector voltage, quicky putting the transistor into saturation and maximising the current thru the coil.
When the inductor is saturated, the field ceases to expand and so the voltage that was being generated in the secondary stops; the voltage to the base is not held so high, and the transistor drops out of saturation. This reduces the current thru the primary, and the inductor's field starts to collapse. The collapsing field generates a voltage in the primary, which boosts the voltage to the LED. The collapsing field also generates a voltage in the secondary, but this time the reverse of the voltage generated earlier; this time the generated voltage pushes the transistor's base toward the emmitter's voltage and further reduces the current thru the transistor, reducing current thru the coil, collapsing the field faster and pushing the transistor into cutoff until the field has totally collapsed.
At this point the base is no longer being driven toward the emitter voltage, the transistor comes out of cuttoff, and current from the emitter again begins to flow thru the collector and coil. The field begins to build, and we are back to the initial condidtion.
The inductance of the primary coil slows the rise and fall of the current, and is the primary determinant of the frequency at which the circuit will oscillate.
Anyone care to verify or refute?