The "surge" that the electric utility was referring to is the demand charge:
The kWh charge (consumption) is the measurement of the amount of energy the building uses over the given period of time. The kW charge (demand) represents the amount of energy consumed at a single point in time. An intuitive way to visualize this is through the car speedometer/odometer metaphor. The rate at which you are using electricity (kW) is comparable to the speed you are driving the car (speedometer). Your actual consumption (kWh) is similar to the total distance in miles that is driven by the car over a given time period (odometer).
In the U.S. most utilities use a rolling 15 minute demand window. The meter averages the demand within each 15 minute chunk of time, and the chunk with the highest average is used to calculate the demand charge. The demand, time, and day all factor in to the amount of the bill.
The day and time matters because the cost of generating and delivering electricity changes. Continuing with the car analogy, certain days and times are like rush hour traffic -- when everyone is trying to use power. The transmission and distribution system (the wires that carry electricity) have physical limits, like lanes of a highway. Demand charges create an incentive for power users to shift the days and times when they use power. Most businesses tend to open around the same time, which is why your employer had a high demand charge. Waiting 15 minutes lowered the average and reduced those charges.
Here's a rough illustration:
Let's assume the area under the two curves (energy, in kWh) is the same. But the red curve (turning all appliances on at the same time) has one big spike in power demand (kW). The blue curve (spacing out when you turn appliances on) has two smaller demand spikes. Since you're only billed based on the single largest spike in the month, the demand charge for the second case will be half of the first case.
Demand charges are still in use today and have mostly increased. Looking at your profile, I see that you're in the Twin Cities, Minnesota, where your utility is likely Xcel. Xcel publishes rate cards which give an overview of the different rates available for homes and businesses. As is typical across the U.S., there is no demand charge for homes.
For businesses, the per kW demand charge is $14.79 in the summer, and $10.79 the rest of the year. There are optional time-of-day rates that limit this demand charge to weekdays between 9am and 9pm, with other charges that apply to peaks at night and on weekends.