Local expression of vector field along a curve In Chapter 2 of Do Carmo's Riemannian Geometry, the following is stated:


Question: In the expression for $V$, why is $X_j=X_j(c(t))$?
 A: The vector field $V$ is defined along $c$. In local coordinates, the restrictions to $c$ of the coordinate vector fields $X_j$ form a basis for the vector fields along $c$, so that each vector field along $V$ is a $C^\infty$-linear combination of the $X_j$ in the local coordinate chart $(\mathbf{x},U)$. When we choose a parametrization for $c$, we have a parametrization for the local coordinate expression of $V$ by pulling back to $\Bbb{R}$:
$$ V(t) = \sum_j v^j(t)X_j(c(t)) $$
tl;dr: The vector field $X_j$ is defined in $U\subseteq M$ so its pullback to $\Bbb{R}$ is $X_j\circ c$. It is the smooth manifold equivalent of a frame field  on a curve from your undergraduate curves-and-surfaces class.

There's some bundle chicanery going on here behind the scenes. The following  diagram chase is as much for my own edification as to answer the question.
We start with a curve $c: L\to M$ where $c$ is a smooth map embedding the smooth $1$-manifold $L$ into $M$, our smooth $n$-manifold. A vector field "along $c$" is a section of the pullback of $TM$ by $c$, which makes the following diagram commute:
$$\require{AMScd}
\begin{CD}
c^*TM @>{}>> TM\\
@VVV @VVV \\
L @>{c}>> M
\end{CD}$$
When we write a vector field in local coordinates, we're really pulling back sections of $TM\to M$ by a local coordinate map:
$$\require{AMScd}
\begin{CD}
\mathbf{x}^*TM @>{}>> TM\\
@VVV @VVV \\
U @>{\mathbf{x}}>> M
\end{CD}$$
In particular, the vector fields $X_j$ are "really" canonical sections of $\mathbf{x}^*TM\to U$ forming a smooth pointwise basis for each fiber. If $V$ is a field defined on $M$, then its local coordinate expression is the pulled-back section $\mathbf{x}^*V: U\to \mathbf{x}^*TM$ when expanded in the basis $X_j$.
A smooth parametrization of $c$ is a choice of local coordinates $\Bbb{R}\to L$ composed with the map $c$ and yields a third commutative diagram:
$$\require{AMScd}
\begin{CD}
\phi^*(c^*TM) @>{}>> c^*TM \\
@VVV @VVV \\
\Bbb{R} @>{\phi}>> L
\end{CD}$$
Now let's put our three diagrams together to see what happens when we take a vector field $V$ on $c$, that is a section $V: L\to c^*TM$, and write it in local coordinates after restricting to a single component of $c(L)\cap U$.
$$\require{AMScd}
\begin{CD}
\phi^*(c^*TM) @>{}>> c^*TM @>{}>> TM @<{}<< \mathbf{x}^*TM \\
@VVV @VVV @VVV @VVV \\
\Bbb{R} @>{\phi}>> L @>{c}>> M @<{\mathbf{x}}<< U
\end{CD}$$
By choosing a parametrization, we pull back $V$ to a smooth section $\phi^*V: \Bbb{R}\to \phi^*(c^*TM)$ of the parametrized bundle. By restricting to a single component of $c(L)\cap U$, we compose with $\mathbf{x}^{-1}$ to get a local coordinate expression $(\mathbf{x}^{-1}\circ c\circ \phi)(t) = (x_1(t), x_2(t), ..., x_n(t))$.
Finally, we pull back the local coordinate bundle $\mathbf{x}^*TM$ and its canonical basis $X_j$ along the map $\mathbf{x}^{-1}\circ c\circ\phi$ to a vector bundle over $\Bbb{R}$ which is bundle-isomorphic to $c^*TM\to L$ and the image of $V$ under this isomorphism is the local coordinate expression of $V$.
There's probably some category theoretic term encapsulating and abstracting this entire process I just haltingly described, but I'm not a category theorist :) 
