$\newcommand{\Reals}{\mathbf{R}}\newcommand{\dd}{\partial}\newcommand{\Basis}{\mathbf{e}}\newcommand{\Brak}[1]{\left\langle #1\right\rangle}\newcommand{\Del}{\nabla}$Let $U \subset \Reals^{n}$ be a non-empty open set, and let $(\Basis_{j})_{j=1}^{n}$ be an arbitrary smooth frame, i.e., an ordered $n$-tuple of smooth vector fields whose values at each point $p$ of $U$ are linearly independent, i.e., are a basis for $T_{p}U \simeq \Reals^{n}$.
A connection $\Del$ in $U$ may be viewed as an axiomization of directional differentiation: If $X$ and $Y$ are smooth vector fields in $U$, then $\Del_{X}Y$, the (covariant) derivative of $Y$ along $X$, is a smooth vector field in $U$ such that for every smooth, real-valued function $f$ in $U$,
$\Del_{fX}Y = f(\Del_{X}Y)$; the value $(\Del_{X}Y)(p)$ depends linearly on, and only on, the value $X(p)$, not on the behavior of $X$ in a neighborhood of $p$.
$\Del_{X}(fY) = (Xf) Y + f(\Del_{X}Y)$; the derivative obeys the Leibniz rule in $Y$.
Writing $X = X^{i} \Basis_{i}$ and $Y = Y^{j} \Basis_{j}$, successive application of the preceding axioms gives
\begin{align*}
\Del_{X}Y &= \sum_{i=1}^{n} X^{i} \Del_{\Basis_{i}} Y \\
&= \sum_{i=1}^{n} X^{i} \biggl[\sum_{j=1}^{n} \bigl((\Basis_{i} Y^{j}) \Basis_{j} + Y^{j} \Del_{\Basis_{i}} \Basis_{j}\bigr)\biggr] \\
&= \sum_{i,j=1}^{n} \biggl[X^{i} (\Basis_{i} Y^{j}) \Basis_{j} + X^{i} Y^{j} \Del_{\Basis_{i}} \Basis_{j}\biggr].
\end{align*}
The derivative $(\Basis_{i}Y^{j})$ of a function along a vector field is determined by the smooth structure. By contrast, the covariant derivatives $\Del_{\Basis_{i}} \Basis_{j}$ are extra structure.
The key idea is this: Since $\Del_{\Basis_{i}} \Basis_{j}$ is a vector field in $U$, it can be expressed uniquely as a linear combination of the frame fields:
$$
\Del_{\Basis_{i}} \Basis_{j} = \sum_{k=1}^{n} \Gamma_{ij}^{k} \Basis_{k}.
$$
The usual geometric parsing of this equation is: If at the point $p$ the vector field $\Basis_{j}$ is differentiated in the direction of $\Basis_{i}(p)$, the Christoffel symbol $\Gamma_{ij}^{k}$ is the $\Basis_{k}(p)$-component of the derivative.
Equivalently, moving the frame $(\Basis_{j})_{j=1}^{n}$ in the direction of the tangent vector $\Basis_{i}(p)$ infinitesimally deforms the frame by the linear transformation with matrix $A_{j}^{k} = \Gamma_{ij}^{k}$.
When $U$ has a Riemannian metric and the frame field $(\Basis_{j})_{j=1}^{n}$ is orthonormal with respect to the metric, each Christoffel symbol can be interpreted as an angular velocity: $\Gamma_{ij}^{k}$ is the rate at which $\Basis_{j}$ rotates toward $\Basis_{k}$ as the frame is evaluated along a curve through $p$ with velocity $\Basis_{i}$.
Because the covariant derivative $\Del_{X}Y$ is linear over smooth functions in $X$, the map $X \mapsto \Del_{X}Y$ (with $Y$ fixed) may be viewed as a vector-valued $1$-form, and the association $X \mapsto \Del_{X}$ may be viewed as a $1$-form taking values in the bundle of endomorphisms of $TU$.
Locally, such a gadget is represented by a matrix-valued $1$-form $(\theta_{j}^{k})$, the connection form, which satisfies
$$
\theta_{j}^{k}(\Basis_{i}) = \Gamma_{ij}^{k}.
$$
Barrett O'Neill's Elementary Differential Geometry is an accessible account of this viewpoint. Particularly, he notes the importance of measuring the rate of change of a frame along a curve in terms of the frame itself.
