Once you figure out how to move the particles in their 3D space you can figure out the range of values you need to set to get the movement you want. Your values depend on how much you want the speed to increase. In this case the groups of 3 keyframes for Wind Z go from zero to -6000. Size over time may help keep the particles from getting too big as they approach the camera. Here's how you would set that up to experiment with the range of values you would need sound keys to generate. One way to do that using Particular would be to animate the speed of the wind in the z direction with the default camera.
Your example video does not show an increase in the number of particles, it shows a change in speed towards the camera. The visual perception of motion is delayed compared with the keyframes so when you are using sound keys to generate the keyframes you'll have to shift some things around in time so the movement matches the beat. There are remnants of the first explosion on the outside, then the second explosion a little bigger and closer to the emitter, then the third explosion is just getting started. The results look something like fireworks going off. Every time there is a group of 3 keyframes a little explosion happens, the particles get bigger very quickly then the size goes down. The size over time is animated by the curve you see in the effects control panel. The sets of 3 keyframes go from zero and 1000 and then back to zero.
Something like this would get you started on seeing how animating the number of particles per second makes them behave. Just poking around with some values and some properties and running previews is pretty much a waste of time. Start by playing with the controls and some keyframes to figure out what approach you want to take before you try and use Sound Keys to control the animation. Once a particle is emitted and started down it's path you can only influence that path, the size and the color of the particle over its lifetime. You can animate gravity, wind, but you cannot increase the number of particles after they have left the emitter. Once the particle has been emitted it is moving with a certain speed. There is an emitter which has a defined size and position, there is the particle, and then there is particle velocity. You have to start by understanding how particle systems work.