Silent shooting mode normally (*please see the end of the answer for a notable exception) does not affect Image Quality in any way. Rather, it affects the way your 5D mkIII cycles the mirror and shutter curtain for each shot you take.
When shooting via the viewfinder:
The Silent mode single option uses a slower speed to move the mirror up out of the light box. The first curtain opens, and then the second curtain closes as normal. But then, nothing else will happen until you release the full press of the shutter button. When you do, the mirror will drop back down, again at a reduced rate, and the shutter curtains will reset. This is very useful when shooting tennis or golf. You can expose a frame virtually silently at the height of a competitor's back swing and hold the shutter down until they have struck the ball and then release the button and allow the mechanisms to reset.
The silent mode continuous option is similar, but goes ahead and allows the mirror and shutter mechanisms to reset at a slower and quieter rate. This reduces maximum frame rate from 6 fps to 3 fps.
In Live View:
The options are similar, but the mirror stays up as long as you are in Live View and are using Contrast Detection Autofocus or Manual Focus:
Mode 1 opens the first curtain prior to the shot and electronically exposes the sensor from bottom to top (top to bottom of the inverted image). The exposure is ended by the movement of the first curtain from the bottom of the light box back to the top of the light box. This is the same direction the first curtain is moved when it is reset following a still shot taken using the viewfinder. The second curtain never moves throughout the entire Live View - Silent Shooting cycle. For each frame exposed, the only sound is the first curtain closing to end the exposure and then reopening for the next shot.
Mode 2 is a single shot mode. The shot begins just like Mode 1. But once the first curtain closes it doesn't reopen to expose the sensor for Live View until the full press of the shutter button is released.
Disable allows shooting with Canon's Tilt-Shift lenses or when using extension tubes (and presumably any third party lens that has an aperture manually set on the lens). When the shutter button is fully pressed using this setting the first curtain resets to the top of the light box, the sensor is cleared, and the first curtain reopens to begin the exposure. The exposure is ended by the conventional movement of the second curtain from the top to bottom of the light box, the sensor is read out, and the second curtain is reset to the top of the light box while the first curtain remains open in the bottom of the light box.
(*The exception would be when you are using shutter speeds subject to allowing the vibrations produced by the mirror and to a lesser extent the shutter curtains to affect the image. Although it varies based on camera design, most researches that have thoroughly tested such vibrations place shutter speeds between about 1/100 second and 1 second as the most vulnerable to mirror vibration affecting the image. Any shorter exposure time and the image has already been recorded by the time the vibrations reach the parts of the camera that affect the image. Any longer and the vibrations don't last long enough to have much effect in the same way that someone running rapidly across the field of view of a 30 second exposure won't show up in the image. Do note that you'll only ever be able to detect blur caused by mirror vibrations if the camera is mounted on solid footing such as a sturdy tripod, the shutter is released using a wired remote cable, an IR remote, or the self-timer and the subject is also perfectly static.)