Startup circuit 0 current mode
Integrated circuits are powered from a power supply that is turned on at some point in time. Ideally such a power supply when turned on starts at zero volts and ramps up to the specified voltage.

So you have a power supply that is initially zero. If you have a circuit that has a stable operating point at such a voltage it will never leave that operating point. It won't start up.

In your bandgap circuit this operating point is possible. If the current mirrors have zero Vgs no current will flow and all the voltage will drop across drain-source. The diode connected transistors don't turn on, your circuit can't start up.

The diode connected transistor conduct a small current because of the subthreshold current. Therefore the PMOS transistors pull the drain nodes to VDD and the NMOS transistors pull the drain nodes to VSS.

A startup circuit will detect this condition and inject a current to get the circuit going. As soon as the circuit leaves the undesired operating point the startup circuit will be turned off.