Structure of Mach diamonds in supersonic and hypersonic flows (SpaceX raptor engine video of Sept. 28, 2019)

Lots of things to cover here. I'll do my best layman's explanation and point you to a resource where you can get your equations and derivations, especially since the one you found and the graphic is a little erroneous on a few details.

Instead of speed, think instead why does supersonic conditions influence the anatomy of the flow. To accelerate a fluid beyond sonic conditions, we have to do a couple things. Our combustion chamber is a high temperature high(er) density reservoir. As the fluid starts to move through the converging section of the nozzle it starts to accelerate. We know from fluid dynamics when a fluid moves through a reduction in area it increases in velocity and conversely its pressure decreases. This happens all the way up to the throat of the nozzle where we reach a condition called "choked flow". At choked flow conditions the flow rate of the fluid is fixed, pending changes to the chamber pressure, temperature, or size of the nozzle. Why? The fluid has reached sonic (Mach = 1) conditions. If you imagine individual particles bouncing around and hitting each other, they are moving at a speed equivalent the wave of information that caused them to move in the first place - imagine a wave of dominos all falling at the same time simply because the one before it slammed it down so fast it all looked like it happened at the same time. Getting the last domino to fall before the first one is seemingly impossible, right?

Now that we have choked flow conditions we reach a turning point that determines the rest of the flow - the pressure ratio of the receiver to the combustion chamber (reservoir) (P_receiver/P_chamber). If the pressure ratio is above 0.9607, then the flow will return to subsonic conditions. This flow readily has the ability to adjust to the conditions of the receiver. However if 0.9707 > p_ratio > 0.436 then the fluid will accelerate past Mach = 1. Because the pressure of the receiver is so much lower than the pressure of the combustion chamber, the density and pressure of the fluid plummet and drive the velocity of the fluid incredibly high.

In this 0.9607 > p_ratio > 0.436 operating condition, the flow only travels Mach > 1 in the diverging section of the nozzle. The pressure ratio is not low enough to drive supersonic conditions outside of the nozzle and a standing normal shock forms. The shock is a rapid inefficient compression process which allows flow adjustment from imposed pressure conditions on the flow. Flow is always subsonic after a standing normal shock. At p_ratio 0.436 the nozzle occurs exactly at the nozzle exit, and as the ratio goes up the shock moves farther and farther up into the nozzle (this is really bad).

Now lets say we have a pressure ratio lower than the one discussed, like 0.436 > p_ratio > 0.0643. That standing normal shock we talked about starts to turn into an oblique shock (shock at an angle to the flow). There are still ridiculous pressure differences between the flow and the receiver, so this shock still needs to exist to compensate for these discontinuities. Now we can talk about your picture.

(These pressure ratios are for an area ratio of 2.494)

The biggest problem with the graphic posted is that it shows the flow smoothly turning before and after shocks. This is not the case. Only a deflection across an oblique shock and a turn from a boundary can cause a supersonic flow to change direction.

SO. First thing that happens is the flow (green line) is incredibly low pressure and moving fast. An oblique shock forms to manage the discontinuity between the flow and the ambient. The flow is directed inwards away from the normal of the shock (red line) and amazingly, this flow is at the pressure of the receiver. This flow is again deflected by another oblique shock and is traveling parallel to our flow path and is compressed again. Now remember we can't have discontinuities between our flow and the ambient. Well, our ambient is a boundary! This flow is a Prandtl - Meyer expansion fan (blue lines) and causes another increase in Mach number, is deflected away from the flow, and decreases to the atmospheric pressure! Amazingly enough after crossing enough Prandtl - Meyer expansion fans the flow is further accelerated, deflected inwards again, and pressure decreases - starting the cycle all over again.

The way to think about it is that oblique shocks and prandtl - meyer expansions are natures way of resolving the discontinuities between the flow and the receiver.

As for the disks - I can't say I know exactly what is going on there, but my guess is that the oblique shocks are deflected so far away that a normal shock occurs - giving us a a strong compression process and a flow that is subsonic in that region. Looking at photos of the disks from vehicles like the space shuttle, it looks like they eventually converge into points as they do in the Raptor photo you provided.

Let me know if you have follow up questions and I'll do the best I can.

The book you want to look up is Gas Dynamics by Oscar and Biblarz and reference chapters 5,6,7, and 8. There may or may not be pdfs readily found on google and it derives quite systematically all the fundamental equations that govern nozzle operation, normal shocks, oblique shocks, and the like. Surprisingly readable, not super dense.