Why could resilient channel improve the sound insulation drywall?Are there some measurement? This post says

Resilient channel is a thin metal channel that is intelligently designed to substantially improve the sound insulation of drywall, sheetrock, plasterboard walls and ceilings.

Why could resilient channel improve the sound insulation drywall? Are there some kind of measurement to evalute the contribution of resilient channel to sound insulation.
 A: Let's check the illustration...

Notice the "resilient channel" which is the horizontal metal profile is only screwed into the wood stud on one side, so it is free to flap around.
So this is mass-spring system.

The mass is the drywall, the "resilient channel" is the spring, and the grey bar the spring is attached to on the illustration is the wood stud in the wall.
Say your neighbor is banging on the walls. The studs vibrate. If drywall is directly screwed on studs then it will vibrate along with the studs and let noise into the room. What the spring does is decouple drywall from stud in an attempt to reduce the amount of vibrations that get into the drywall. 
It works both ways, if you listen to loud music this will make the drywall vibrate, and the resilient channel will prevent this vibration from propagating into the studs and the whole structure.
This is called "decoupling" because it allows both the drywall and studs to vibrate independently.
If you want math, check harmonic oscillator. An important physics notion is resonance frequency, which depends on masses at both ends of the spring, and the stiffness of your spring. This system will attenuate noise above resonance frequency, but not below.
So, basically you want the spring to be as soft as possible for lowest resonance frequency (best soundproofing), but it still has to hold the drywall in place, even if you hang stuff from the wall. So spring stiffness will be a compromise between soundproofing and structural integrity.
Adding mass (for example double layers of drywall) will also lower resonance frequency and decrease sound transmission. The force transmitted to the drywall depends on spring stiffness, but how much the drywall moves (vibrates) depends on its mass by F=m.a.
This looks like a good system, especially for ceilings. It seems to be thinner than the decoupling "suspenders" I've seen that are used to hang standard steel drywall channel, so maybe a good choice if you don't want to drop your ceiling too much.
Note if you want to hang heavy heavy stuff on your wall, like kitchen cabinets, screwing them into the studs will compress the channel against the studs and defeat the decoupling.
A: I haven't come across a resilient channel (but to be honest I am not very active on sound insulation acoustic consultancy), but from the site you cited and some more like this, it seems that what people claim that resilient channels achieve is to isolate the wood/metal frame of the partition (wall, ceiling or floor) from its two sides, so that vibrations induced by sound (mainly airborne, most probably). In this way, you do not allow vibrations from one side of the partition to reach the other side and re-radiate into adjacent rooms.
As I said I haven't encountered one yet so I do not know if you can get any "numbers" regarding its insulating capabilities. On the other hand, I am not sure how someone could manage to provide any "decibel values". It should somehow follow some standardised way of measuring (there are standard constructions for sound insulation measurements. For more info, you can check ISO 10140-5:2010), and I am not sure there's a standard for this kind of materials (I may be mistaken though).
Maybe the manufacturer could provide some more info along with values for the properties of the material that could be used in predictions and modelling. This whole thing resembles the vibration isolators (have a look here and here) used for what their name implies... vibration isolation. So, maybe the manufacturer provides similar values for these resilient channels. No matter what, contacting the manufacturer is a good start because they know best in what way their products can and are meant to be used.

UPDATE
Just integrating JMac's comment here. It seems that the manufacturers are not "lying" about it. According to JMac's comment:
"It does mention a 3-5 "(or more)" point increase on the Sound Transmission Class, which does seem to be an actual system for measuring (I just found an ASTM page (link - https://www.astm.org/Standards/E413.htm -) that referenced it), so assuming they aren't just lying about that, they did give some rough numbers."
