How does a solar garden LED work with just 1 AAA battery?
There are two pieces that work together in order to light the LED.

The first is an IC such as the QX5251. This is a solar battery charger and LED driver in a TO-94 package. It rectifies the solar input as well as detects when it is dark in order to activate the LED.

The second is an inductor which the driver uses to boost the voltage from the battery high enough to light the LED. The driver uses the same mechanism as a boost regulator but without any of the filtering.

The result is an LED that is imperceptibly flickering. This flickering may interfere with  your vision attempts, so do plenty of testing before deploying it in the wild.