A typical way of an incandescent bulb to burn out is with a bright blueish flash and often an audible sound and (very rarely) an explosion of the bulb glass.
Since a filament breaking in a gas-filled bulb can form an electric arc, which may spread between the terminals and draw very heavy current, intentionally thin lead-in wires or more elaborate protection devices are therefore often used as fuses built into the light bulb.
But it's unclear why an arc would spread and cause very high currents.
Why would it not burn out silently instead?
Best Answer
TL;DR The electric arc may conduct much higher current than the filament, this is not normal mode of bulb operation so you can see a flash and maybe even have the bulb exploded.
Typical incandescent bulb failure develops while current is flowing through the filament. The filament is a piece of wire about half a meter long coiled such it forms a filament which is about 20 millimeters long. The filament wire is therefore very thin and long and its resistance is relatively high. The filament is attached to the upright supply wires which are not coiled - they are pieces of wire much thicker than that of the filament, their resistance is relatively low.
While the bulb is working current flows through the supply wires and the filament. Filament resistance is what limits the maximum current.
If the filament breaks while current is flowing through it then this situation is identical to starting an arc with electric arc welder. The filament breaks while current is flowing though it, its ends move from one another and chances are an electric arc starts.
While the filament was intact the maximum current was initially limited by the filament resistance and there was no way for the current to flow other than through the entire length of the filament. If an electric arc starts then this limitation is no longer true - current can flow through the ionized gas and the length the arc can span is limited by the power supply voltage. This allows the arc ends to travel from the broken filament ends to the supply wires and no longer limit its current by the filament resistance. Once the arc starts it ionizes the surrounding gas and so that gas can conduct current too which allows for much higher currents than the one initially flowing through the thin filament. The current is now limited by the supply wires resistance.
This causes rather intensive arc which produces a bright flash and sometimes makes the bulb explode.
This is why you need a fuse inside the light bulb (near the socket). If arc develops then current through the fuse gets much higher and the fuse is melted and then the circuit is broken and the arc stops. If the fuse fails to melt fast enough you may have a circuit breaker tripped.