Digital cameras work with electronic light sensors (CCD or CMOS), which capture incoming light quantity for each of the three primary colors (red, green and blue) for each image "dot" or pixel. Those sensors are designed to differentiate between thousands of intensity (or illumination) levels for each of those colors. In order to do this, reference values (for black and white) must be very precise, and the analogue-to-digital converter (ADC) has to output with very high resolution (up to 16 bits per channel or color).
The design of those sensors optimizes the signal to noise ratio to obtain the right reading, so that the captured color is "like" the real one. Sensor manufacturers design those sensors to work with sensitivities around the original ISO 100 values - but you may find cameras with "native" sensitivities from ISO 80 to ISO 200.
From that starting point, the advantage of electronic circuits is the ability to modify the gain of the sensor (by amplifying the signal and / or shifting the ADC reading). Easiest procedure involves binary calculation, so duplicating the sensitivity is quite direct. Then, we can repeat this process until we reach saturation levels in the signals. This is, we will go to 200, 400, 800, 1,600, 3,200... each camera will have a given ISO limit.
When we adjust the ISO setting, we are modifying the gain of the sensor system. This is equivalent to work with higher film sensitivities. And similarly to film photography, a high ISO setting will allow for the capture of photographies in low light, or rather at high speeds.
Drawback from increasing the sensor gain is, as in any electronic device, the generation of electronic noise (the signal-to-noise ratio gets worse), which typically will produce wrong color pixels in the final image - something visible in even, uniform areas of the pictures. This effect cannot be compared with traditional film grain, which in some occasions could be considered to be aesthetically right...
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