My strands of LED Christmas lights are not polarized, and I don't understand how that works
The ac portion of the led string is not polarized. Typically a full wave diode bridge, which is polarity independent, will be part of the string to convert ac to dc and a regulator circuit after that to regulate it down to a stable dc voltage. Leds by their nature are directional which is why their are polarized as you say. Depending on the number of lights, the strings are subdivided into sections so pulling one bulb would only affect one section. There is also a pass through connection between sections for the AC connection.