I think what's happening here is that you are reading "the reduction of exposure corresponding to use of higher sensitivities generally leads to reduced image quality", and then, very logically, thinking: okay, so, reduced exposure = reduced image quality, so clearly I need increased exposure....

This is, in fact, completely right: see this answer on digital ISO for some more explanation.

That link and the Wikipedia article are both saying the same thing. There are three factors that you can change when photoraphing given scene (with a set amount of light): aperture, shutter speed, and ISO (sensitivity or gain). Increasing the first two increases the actual light that reaches the sensor or film: a longer shutter speed lets light in for more time, and a wider aperture lets more light in at once.

The ISO, however, is how quickly the sensor or film responds to that light. If you increase it but keep the other factors the same, the too sensitive ISO means that the image is overexposed. So, that's why you must reduce them. And, because physics can't be beat, less light means less data, and less data means reduced image quality. (Fortunately, with modern equipment, this reduction might be hard to perceive until the ISO gets very high.)

This is all extra confusing because people use the word "exposure" in multiple different ways, all slightly contradictory. See this answer where I go into detail, but, basically: often people include aperture, shutter time, and ISO, and possibly even post-processing into "exposure factors" — with "exposure" basically meaning the resulting overall brightness. This is a perfectly fine and common definition, but another equally correct* definition only considers the amount of light allowed to hit the sensor — just the shutter time and aperture, and that's the definition used in the Wikipedia quote.

So, when I say that you're completely right, I mean that you need to have increased exposure in the latter sense: just more light. Unfortunately, what the Wikipedia article is saying is also completely right: you can't have that with a higher ISO without changing the exposure in the other sense, rendering your output too bright.

(The only way out is to add more light: change the time of day, or add flash or other artificial light. But that's a whole different story.)

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* (possibly, pedantically more correct, but... English is inherently a sloppy language)