As far as I'm aware, a vector field is defined as a function $F:U\rightarrow\mathbb{R}^n$ for some subset $U\subset\mathbb{R}^n$. However, it is not explained why the function must map elements in $\mathbb{R}^n$ to vectors in the same space. Why can't a vector field be defined as $F:\mathbb{R}^n\rightarrow\mathbb{R}^m$?
I've discussed this with others and the reason seems to be something along the lines of "the concepts we've defined don't make sense for such vector fields." For example, the concept of divergence defined as
$$ \nabla\cdot F = \sum_{i = 1}^n{\frac{\partial F_i}{\partial x_i}} $$
would no longer be well defined for $m \neq n$, since $F_i$ wouldn't be present for $i > m$ and $x_i$ wouldn't be present for $i > n$. Additionally, path integrals defined as
$$ \int_\gamma{F\mathrm{d}t} = \int_a^b{\langle F(\gamma(t)), \gamma'(t)\rangle\mathrm{d}t} $$
wouldn't be well defined, since the standard scalar product only operates on vectors with equal dimension.
But these are just definitions. All of these are concepts we defined, and we could define them to only apply to certain functions e.g. only the ones that map elements from $U\subset\mathbb{R}^n$ to $\mathbb{R}^n$. We have various classes for functions (continuous, differentiable, injective, etc.) and in theorems we require certain properties and only make statements about objects with the desired properties. Why is this not the case for vector fields?
You could define a vector field to take vectors in some completely different space. If you do, you get the the general notion of vector bundles. They don't even require the underlying base space to be a vector space, only that it is a manifold; the Earth mapped with the temperature and air pressure at each point is a standard example of a vector field (more commonly called a section) $S^2\to\Bbb R^2$, but taking just the temperature gives us a vector field $S^2\to \Bbb R$.
However, in many cases in calculus and beyond there is one particular vector bundle we are interested in, and that's the so-called tangent bundle. Here, the space of possible vectors at any point is directly inherited from the structure of the base space itself: At each point $p$, the space of possible vectors at $p$ is given by all possible first derivatives that curves going through $p$ can have at $p$.
So if our base space is $\Bbb R^n$, then at any point $p$, the space of possible derivatives that a curve going through $p$ can have at $p$ is $\Bbb R^n$. Formally and rigorously setting up a vector space structure (in particular vector addition) on this "space of derivatives at $p$" takes a bit of work, and I believe intro courses to differential geometry typically spend about a full lesson on it.