According to Bongiovi, the chip "adapts intelligently to the music to give even cheap speakers a full, robust sound and compensate for the deficiencies of the listening space". As far as technical explanations go, in the original article you can find only this: "it can be described as a very sophisticated equalizer".
Any audio enthusiast knows that equalizers have their use in high-end hi-fi, but the good ones come at a very high price. If Bongiovi managed to reach great levels of performance with a cheap chip, then this is indeed newsworthy.
But the claims that the chip will make the sound from cheap computer speakers "rich", "clear" and "free of distortion" seem very suspicious to me. Because, when it comes to cheap, small computer speakers, they are not limited by the electronics - they are limited by the physical properties of the case and the membrane and the size of the membrane. In other words, no amount of magic will make them sound good. A bit better, maybe; great - never.
Another use of the chip that is mentioned is car stereo, and it makes much more sense to me. If the equalizer is preprogrammed for the exact topology of a certain car model and then sold for that exact model, the results might be really nice. The price for the car audio system is said to be $700 - $1000, depending on the model (and we guess this doesn't include the speakers).