Note that the holiday lights are resistive loads
If resistors are connected in series, the same current will flow through them. Since the holiday lights are connected in series, the same current flows through them, and if a light goes off, the other lights also go off.
But there are special lamps used to short out a lamp when the potential difference across it increases to the line voltage, The essence of this special lamp is to ensure that even when one lamp fails, the entire holidays lights do not go off.
After many bulbs have failed, the resistive loads reduce, that is, the overall resistance increases, leading to an increase in the amount of currents passing through the bulb. The abnormally high amount of current that is now passing through these bulbs is enough to blow their fuses