Figure 28 of your datasheet shows how startup is delayed until the SS/TR pin rises above 800mV. It has an internal current source and recommends the delay capacitor have a value less than 470nF and greater than 470pF. In many ways it is just another comparator to judge if the IC should be ON or OFF-or delayed.
Soft-start can help prevent overshoot of the output voltage compared to IC's that are ON at full voltage very fast-too fast to control any overshoot which could harm some types of logic IC's.
This is a quote from the datasheet about softstart:
7.3.7 Soft-Start/Tracking Pin (SS/TR) The TPS54341 device effectively uses the lower voltage of the internal voltage reference
or the SS/TR pin voltage as the reference voltage of the power-supply
and regulates the output accordingly. A capacitor on the SS/TR pin to
ground implements a soft-start time. The TPS54341 device has an
internal pullup-current source of 1.7 μA that charges the external
soft-start capacitor. The calculations for the soft-start time (10% to
90%) are shown in Equation 5. The voltage reference (VREF) is 0.8 V
and the soft-start current (ISS) is 1.7 μA. The soft-start capacitor
should remain lower than 0.47 μF and greater than 0.47 nF.
It often works by limiting the maximum pulse width of the output MOSFET's which would normally start at maximum pulse width with a heavy surge of current and a possible brief overshoot of the output voltage. Soft start puts a throttle on that so it starts at a narrow pulse width but quickly allows the pulse to widen to its normal width for a given load, so the loop stabilizes quickly. That is why the soft-start capacitor has a narrow range of values. Soft-start is a brief choking of the MOSFET pulse width-no more than that.