According to this: https://en.wikipedia.org/wiki/Gornergrat much of the water ends up in the Grande Dixence dam.
Located in the Pennine Alps, the Matter Valley is drained by the Matter Vispa, a tributary of the Rhone. The valley itself ends at Stalden where it meets the Saas Valley. The resulting Visp Valley continues for a few kilometres until it reaches the town of Visp on the young river Rhone. The valley starts between the high summits south of Zermatt (Monte Rosa, Matterhorn) on the border with Italy. The upper side is glaciated, the second largest glacier of the Alps, the Gorner Glacier lies at the foot of Monte Rosa (4,634 m), while the Zmutt Glacier lies at the foot of the Matterhorn (4,478 m). Around the village of Randa are located the Weisshorn (4,505 m) and the Dom (4,545 m). The difference of height between the talweg and the summits on both side reaches over 3 km. The total length of the valley is about 40 km.
Sourced in the mountain basin that surrounds Zermatt, this river flows through town then down through the valley that, if you remember, is overlooked by Oh Wow point.