It would be wrong to think that increasing ISO results in no "physical" change in the camera at all. The problem with ISO is that people often call it sensitivity. That is really a misnomer...sensitivity is a fixed attribute of any given sensor, and it cannot be changed.
Sensitivity is really more synonymous with the quantum efficiency of the photodiodes, factoring in the percentage of light filtered out by the IR cutoff filter, low-passe filter, and color filter array. Generally speaking, most DSLR's these days actually have a 13-18% "sensitivity" to light...that means that only about 13-18% of the light that passes through the lens actually reaches the photodiode and actually releases an electron. Approximately 60% of light or more is filtered by the filter stack and CFA, and the quantum efficiency of photodiodes (the rate of photon strikes to electron release) in moderns sensors ranges from ~45% to ~60%.
ISO is really just an instruction to the camera's electronics to change how the electronic charge, the analog signal, stored in the sensor is amplified to produce a proper exposure. In that sense, there IS a "physical" change in what is actually happening to the image signal within the electronics of the sensor. A certain gain is applied to the original signal in the sensor upon readout. Increasing ISO changes this gain, resulting in greater and greater amplification of the signal.
Your question is, does increasing ISO matter, and does changing ISO affect the RAW? The answer is YES and YES! If you shot everything at ISO 100, and changed "amplification" digitally in post, then your images will be much noisier than if you use the most appropriate setting in your camera. The mechanisms of amplifying the image signal performed by the camera are vastly superior to your basic digital ISO boost with some kind of post processing tool like Lightroom. The sample image from Matt Grum that you referenced from the original post should be an ideal example of that. Notice how much worse the noise in the digitally boosted ISO 100 image is than the ISO 1600 image? There is a lot of blue color noise, banding patterns are starting to show up, and there is a loss of detail. The camera did a much better job when it was told to use ISO 1600...there is less noise, more detail, sharper detail.
The reason boosting ISO in-camera is better is it works with the original, native signal right off the sensor before any downstream electronics have a chance to introduce additional noise. In a CMOS Image Sensor (CIS), each and every pixel has built-in noise reduction circuitry (CDS, correlated double sampling...this measures the dark-current charge in the pixel at "reset time", and memorizes it so it can be subtracted at readout) as well as a built-in amplifier. When each column of pixels is read out, the charge of the pixel is first denoised by the CDS circuitry, and that "clean" charge is then directly amplified before being sent to the readout circuitry sent off-die. Analog-to-Digital Conversion, or ADC, occurs off the sensor die in a DSP chip in most cameras (there are some exceptions, more in a moment).
ADCs are usually moderately parallel, there may be eight, sixteen, maybe more in a given camera. Despite being parallel, each one must still process hundreds of thousands if not millions of pixels in a fraction of a second. That requires a high operating frequency, which has the tendency to introduce additional noise. This is the primary source of color and banding noise in most DSLRs that exhibit such. The ISO 100 image that was boosted in post is also boosting this additional post-read noise that is introduced downstream from the sensor.
By increasing ISO in-camera, you amplify the image signal directly, and any additional downstream contributors to noise only affect the lower end of the signal. This preserves the image to electronic noise signal ratio. There is an additional contributor to noise that has nothing to do with the electronics. The random nature of light itself results in a Poisson distribution of photon strikes. With less total light striking the sensor, Poisson noise will be higher. If you had a noiseless sensor, one which did not introduce any electronic noise of it's own at all...then using ISO 1600 would be the same as using ISO 100 and boosting the exposure by four stops in post. The amount of noise in the two images would be identical, and it would all be noise that results from the random physical nature of light.
There is one sensor on the market today that is nearly noiseless. The Sony Exmor sensor uses a highly advanced, column-parallel, ON-DIE Digital ADC/CDS readout design. Unlike most sensors, which maintain an analog signal throughout a pipeline from sensor to DSP (until just after ADC), Exmor performs both CDS and ADC on-die, and in a digital manner. Rather than each pixel having analog CDS circuitry to measure dark current per-pixel, Exmor performs a reset-read, that reset read is immediately converted into digital, and stores the entire sensors "dark current" charge in a virtual image of negative values. When an exposure is made, the image signal is read out, converted into digital, and the negative reset image is applied to the positive exposure image.
Since there is one ADC per column in Exmor, rather than one ADC per dozens of columns, they can operate at a lower frequency. Between using digital CDS, per-column ADC, and lower frequency components, Exmor introduces nearly zero noise, does not introduce any visible banding or pattern noise, and for all intents and purposes could be considered a "noiseless" sensor. There is still some noise, and sufficiently boosting an exposure in post will ultimately result in that noise becoming visible. One could, however, take a photo at ISO 100, lift it by four stops, and have it look just about as good as a photo taken at ISO 1600. As a matter of fact, in the case of Exmor...that is EXACTLY the case! All "amplification" in Exmor is digital by nature, although the sensor electronics tend to be better at it than manually lifting exposure in post, by a small margin.
It is also important to realize that the increase in ISO itself does not actually add noise itself. ISO is not a source of noise! Assuming a noiseless sensor, if you expose a static scene such that you achieve a proper exposure at ISO 100, and expose the same static scene such that you achieve a proper exposure at ISO 3200, the latter will have more noise. Why, you ask? Poisson noise, more commonly called Photon Shot noise, or noise caused by the random nature of light, is the cause. In the properly-exposed ISO 100 image, you are either using a wider aperture, using a longer shutter, or both. Let's assume for the sake of discussion we are only changing shutter speed, to maintain DOF and get exactly the same scene at both ISO 100 and ISO 3200. The difference in shutter speed is five stops. That is a thirty-two fold difference in the amount of light at the sensor! The more light you have, the less photon shot noise will be apparent...the signal-to-noise ratio (SNR) of the image to it's own natural noise is higher with a properly exposed ISO 100 image, and much lower with a properly exposed ISO 3200 image.
If we use the Nikon D800 (which uses a Sony Exmor sensor), shooting an image underexposed by five stops at ISO 100, and another properly exposed at ISO 3200, and amplify the ISO 100 image, it will be ever so slightly noisier than the ISO 3200 image. It effectively has the same SNR relative to photon shot noise, and will have a very small contribution of read noise as well that will get amplified along with the rest of the image.
Well, you asked to know exactly how ISO worked in a modern digital camera. This is not a complete explanation, and different manufacturers handle certain high ISO settings differently. For example, Canon sensors will only amplify the image signal directly off the sensor up to a certain point, and will then use an additional downstream amplifier between the sensor and the ADC to achieve the top couple of stops (i.e. in a camera that goes to ISO 6400, ISO 1600 is the max "natively amplified" setting, and ISO 3200 and 6400 involve an additional downstream, but still analog, amplification.) "Expanded ISO" settings are also special on most cameras in that all they really are is a digital boost. So any setting called HI, or H1, H2, etc., is not a true ISO setting...it is a fake ISO setting. You could achieve similar or better results by underexposing in RAW at the maximum real ISO setting, and performing the same exposure boost in post.