Yup, that is a ski area. It is called "Nordpark" nowadays, although the locals call it "Seegrube" (or "Gruabn" in the local dialect). The cable car station you can see in this pic is the highest point of this area, the "Hafelekar". The slopes starting there are very steep and unmarked, bearing the danger of avalanches. The fences are indeed intended to prevent those.
The "Seegrube" is below the "Hafelekar" and has more moderate slopes. Especially on weekends it is really crowded up there since it is so close to the city of Innsbruck. It is a 10 minute car drive from the city center to the cable car station and another 10 minutes with the cable car.
Unfortunately, the area is closed until December because our crazy mayor is building a new cable car from the city to the Seegrube.