How does a soft start Power Supply work?.

How does a soft start Power Supply work?.

Not sure what you mean, but a soft start is usually used in a high power power supply that has a capacitor bank on the secondary. A discharged capacitor bank is a dead short the instant a power supply starts. So to prevent a high current surge to the secondary diodes, etc. the primary gets a "soft start", basically a reduced voltage/current for a brief instant to give a limit charge to the capacitor bank.
This usually involves a resistor put in series with the primary, for a fraction of a second. This limits the current inrush. After about a few AC cycles, the resistor is bypassed with a relay, shorting out the resistor. and allowing full current. This lets the capacitor bank charge "a bit", so there is some voltage on it already, and when full power/voltage is restored, there is not so much of a "jolt" of current.
Some soft start devices use NTC thermistors to limit current instead, but those have to be shorted out after a brief period as well, else they will overheat and burn out.
Same for the resistor design, they have to be shorted out within (about) 100ms or the resistor will overheat and burn up. All resistors have a short interval of "surge" capability, again, in the order of 50ms to 100ms with high current before they overheat.
For example, my kilowatt amplifier has a 22 ohm resistor in series with the 240VAC primary, which I short out in about 100ms. In that 100ms, the capacitor bank charges up to about 500-600VDC. Once the relay shorts out the limiting resistor, the transformer gets full power and the capacitor bank then reaches its designed 3600VDC. No issues. At one time the relay failed to close, and the resistor blew up into pieces. A 3A safety fuse in series with the resistor prevents that now. Worst case now, if relay fails, the fuse blows and disables the primary voltage.
Many step start circuits use a relay that senses/uses the primary voltage. As the resistor allows the capacitor bank to charge, the primary voltage starts to increase to the proper voltage (from seeing a "dead short" to some current flowing). Once the voltage gets high enough, the relay the closes and shorts out the current limiting resistor (or thermistor)

A "soft-start" power supply typically is one designed to bring the voltage up relatively slowly when input power is applied, both to eliminate possible overshoot in the output voltage, and reduce the starting current surge at both the input and the load.

I experimented with several 50 watt resistors in the primary winding transformer circuit. I tried 5 ohms and 20 ohms. I tested relay coil it is 582 ohms. I used a 600 ohm 5w resistor in series with the relay. It takes about 1¼ seconds for the 24vdc PS to come up to 24VDC with a 20 ohm 50w resistor. I repaired the mosfet circuit then turned it on/off several times and never blew any more mosfets. Circuit appears to be fixed.

Another source of inrush is the transformer saturating. Depending on where in the mains cycle the transformer is powered, it can saturate for up to 1/4 cycle and there is change in magnetic field, so no back emf. Without a back emf the current is only limited by the winding resistance and it can get very large.

I've had transformers over 1 kW that have very large inrush currents with nothing connected to the secondary.

Using an NTC resistor or fixed resistor with a relay can reduce the inrush considerably.