You might want to look at applied sciences. Weather forecast seems to be a nice example which shows how a rather easy question leads to high dimensional vector spaces.
Suppose you want to predict the temperature for tomorrow. Obviously you need to take today's temperature into account so you start with a function $$f:\mathbb R\rightarrow\mathbb R,~x\mapsto f(x),$$ where $x$ is the current temperature temperature and $f(x)$ your prediction. But there is more than just the current temperature you have to consider. The humidity is important as well, so you modify your function and get $$\tilde{f}:\mathbb R^2\rightarrow\mathbb R,~(x,y)\mapsto f(x,y),$$ where $y$ is the current humidity. Now, the barometric pressure is important as well, so you modify again and get $$\hat{f}:\mathbb R^3\rightarrow\mathbb R,~(x,y,z)\mapsto f(x,y,z),$$ where $z$ is the current barometric pressure. Already this function can't be visualized, as it takes a 4-dimensional coordinate system to graph it. When you now take into account, that there are many more factors to consider (e.g. wind speed, wind direction, amount of rainfall) you easily get a domain with 5,6,7 or more dimensions.
