Proving df is the sum of partial derivatives Wikipedia states:

Since any vector $v$ is a linear combination $\sum v_je_j$ of its components, $df$ is uniquely determined by $df_p(e_j)$ for each $j$ and each $p \in U$, which are just the partial derivatives of $f$ on $U$. Thus $df$ provides a way of encoding the partial derivatives of $f$. It can be decoded by noticing that the coordinates $x_1, x_2, \ldots, x_n$ are themselves functions on $U$, and so define differential 1-forms $dx_1, dx_2, \ldots, dx_n$. Since $\frac{\partial x_i}{\partial{x_j}} = \delta_{ij}$, the Kronecker delta function, it follows that
$$df = \sum_{i=1}^n \frac{\partial f}{\partial x^i} \, dx^i$$

Can anyone elaborate on this? In particular, how do we know what the components of $df$ are? What's the relationship between $x = (x_1, x_2, \ldots, x_n)$ and $p$? Where does the Kronecker function come in?

It might help to eschew the more complicated topic of differential geometry and look at vector calculus since everything locally amounts to doing vector calculus anyways.
If $f$ is a scalar function, then $\mathrm{d}f$ is just the ordinary derivative $\nabla f$. And if, for example, $\left[ \begin{matrix} x \\ y \end{matrix} \right]$ are the coordinates on the plane, it's easy to see that $\mathrm{d}x = [ \begin{matrix} 1 & 0 \end{matrix}]$ and $\mathrm{d}y = [ \begin{matrix} 0 & 1 \end{matrix}]$ are the standard basis vectors, so that
$$\mathrm{d}f = \left[ \begin{matrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \end{matrix} \right]
= \frac{\partial f}{\partial x} \mathrm{d} x + 
\frac{\partial f}{\partial y} \mathrm{d} y$$
(Yes, in coordinates, $\mathrm{d}f$ should be a row vector, whereas vectors representing things like position or direction should be column vectors. The formula $\nabla_v f = \mathrm{d} f \cdot v$ is a product of matrices, not a dot product)
(Also, be aware some people insist that $\nabla f$ is a column vector in coordinates, so that $\mathrm{d}f$ is the transpose of $\nabla f$. I, myself, do not find that a particularly useful convention)
