In what way do battery grips improve performance?

There are multiple camera models, both DSLR and mirrorless, which get improved performance (higher frame rate, smaller shutter lag) from adding a battery grip.

I would like a technical explanation about how an additional battery is able to improve performance. Naïvely, one would expect that an extra battery serves only to provide power for a longer time. We do not see better performance from a smartphone or laptop when connecting a power source (provided that all power saving features are disabled), so what is different about a camera?

Answer

Two batteries in parallel at the same voltage can supply more current than one of them alone can. If we were talking about car or truck engines we would say we have twice as much torque (current) at the same RPM (voltage). Therefore the same load will not cause the same drop in voltage when two of the same batteries are in parallel. This also means a greater load can be put on the two batteries before the voltage will drop by the same amount as a lesser load will cause the single battery's voltage to drop.

Nikon markets it more, but as voltages in the system drops as the batteries use their capacity other camera makers' cameras do the same thing - certain performance parameters are reduced. If you read detailed specs and performance parameters (as compared to marketing numbers) for many high performance bodies carefully you will see that maximum frame rates usually are qualified with something like "... with batteries at 75% or more of capacity."

In the case of Nikon it seems to be the case that they also use the number of batteries and their level of charge to 'throttle' the frame rate via firmware. Only when there are two batteries in the grip and both are above a certain level will the camera use the increased frame rate.

Canon doesn't market the fact, but their cameras' advertised max burst rate is only possible when a single battery has about 75-80% of full capacity. With a grip one can get the same performance as a single full battery when both batteries are at least 50% or higher. Based on the way Canon often handles public relations and reveals information, it's my opinion that they fear what many of their customers would do if they realized the camera slowed down slightly when the battery is below 75%.