If we have a function such as: $\mathbb{R}$ to $\mathbb{R}$, $f(x)=x$, we have a one varible function, that is living in a two dimensional space, because n+n = 1+1 = 2, so, if we have the following function: $f(x,y)= (xy, xy^{2})$, it must be a two dimensional manifold, which exists in $\mathbb{R}^{4}$. How is this possible to visualize?
Why is it that given $f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ n+m = the dimension the manifold is in?
By definition, the graph $\{(x,f(x))\}$ of $f$ is a subset of $\mathbb{R}^n \times \mathbb{R}^m \cong \mathbb{R}^{n+m}$.
As for your other question, it's possible to visualize functions $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ via color wheels.