The problem of getting power to startup equipment is one thing, but there's another cool problem that this article covers but I don't think explained very well.
A power grid is a massive distributed physical system. The energy input must exactly match the energy output at all times.
But what happens when the energy ins and outs are not balanced? The answer is that a balance is found somehow. Physics demands it.
If there is excess power on the grid and no electrical load, that power comes back to the generator (s). The turbines or whatever driving the generators produce more torque than the retarding torque from the generator coils, so they speed up. The AC grid frequency is mostly maintained by the rotating generator speed (3000 rpm for 50 Hz), so that goes up too.
Conversely, if there is excess load and not enough power, electrical drag from the generator coils exceeds the torque from the turbine (or whatever), and the generator slows down. The operator has to burn more fuel, or pull out control rods, or open more water gates, to get the speed back up.
So what is the black start challenge here? You have to go from 0 W to whatever GW the grid normally runs at. Normally when a generator plant is switched onto the grid, that gen represents a small fraction of the total grid power, so the disturbance to the grid is small. But coming back from a black start that's not true. Going from 1 plant online to 2: you could be doubling the power level. This means you have to switch on loads (possibly many km away) at the exact same time you switch in the power. If the disturbance is too much, various equipment will trip off the grid as the AC frequency careens out of control.