This is Proposition 8.16 from Tu's book An Introduction to Manifolds.
For any point $p$ in a Manifold $M$ and any tangent vector $X_p \in T_pM$, there are $\epsilon > 0$ and a smooth curve $c: (-\epsilon, \epsilon) \rightarrow M$ such that $c(0) = p$ and $c'(0) = X_p$.
His proof (in terms of notation, Tu uses the the subscript $*$ to indicate the differential and $'$ to indicate the velocity vector):
Let $(U, \phi) = (U, x^1, \ldots, x^n)$ be a chart centered at $p$; i.e. $\phi(p) = 0 \in \mathbb{R}^n$. Suppose $X_p = \sum a^i \partial/\partial x^i|_p$ at $p$. Let $r^1, \ldots, r^n$ be the standard coordinates on $\mathbb{R}^n$. Then $x^i = r^i \circ \phi$. To find a curve $c$ at $p$ with $c'(0) = X_p$, start with a curve $\alpha$ in $\mathbb{R}^n$ with $\alpha(0) = 0$ and $\alpha'(0) = \sum a^i \partial/\partial r^i|_0$. We then map $\alpha$ to $M$ via $\phi^{-1}$. By Proposition 8.15 (which says the velocity of a curve in local coordinates is $c'(t) = \sum \dot{c}^i(t) \partial/\partial x^i |_{c(t)}$ relative to the basis $\{\partial/\partial x^i|_p\}$), the simplest such $\alpha$ is $$\alpha(t) = (a^1 t, \ldots, a^n t), \quad t \in (\epsilon, \epsilon)$$ where $\epsilon$ is sufficnetly small that $\alpha(t)$ lies in $\phi(U)$. Define $c = \phi^{-1} \circ \alpha: (-\epsilon, \epsilon) \rightarrow M$. Then $$c(0) = \phi^{-1}(\alpha(0)) = \phi^{-1}(0) = p$$ and by Proposition 8.8 $$c'(0) = (\phi^{-1})_*\alpha_*\Big(\frac{d}{dt}\Big|_{t = 0}\Big) = (\phi^{-1})_*\Big(\sum a^i \frac{\partial}{\partial r^i}\Big|_0\Big) = \sum a^i \frac{\partial}{\partial x^i}\Big|_p = X_p.$$
I am stuck on the last line of this proof. Proposition 8.8 says
Let $(U, \phi) = (U, x^1, \ldots, x^n)$ be a chart about a point $p$ in a manifold $M$. Then $$\phi_*\Big(\frac{\partial}{\partial x^i}\Big|_p\Big) = \frac{\partial}{\partial r^i}\Big|_{\phi(p)}.$$
My interpretation of the above proposition is that the differential $\phi_*$ maps the $i$th basis vector of the tangent space of the domain to the $i$th basis vector of the tangent space of $\mathbb{R}^n$. How does he use this to conclude that $$(\phi^{-1})_*\alpha_*\Big(\frac{d}{dt}\Big|_{t = 0}\Big) = (\phi^{-1})_*\Big(\sum a^i \frac{\partial}{\partial r^i}\Big|_0\Big) = \sum a^i \frac{\partial}{\partial x^i}\Big|_p?$$ For example, how does he conclude that $$\alpha_*\Big(\frac{d}{dt}\Big|_{t = 0}\Big) = \sum a^i \frac{\partial}{\partial r^i}\Big|_0$$ and likewise for $(\phi^{-1})_*$? Where are these summations coming from? I'm guessing it comes from a chain-rule type of argument, but this is only my intuition from $\mathbb{R}^n$, and I am not sure if this is correct for an arbitrary manifold.
Thanks to the comments of l4teLearner I was able to figure out this question. The tangent space of the curve $\alpha: \mathbb{R} \rightarrow \mathbb{R}^n$ is $n$-dimensional. This is why the differential $\alpha_*$, a map to and from tangent spaces, outputs a linear combination of $n$ basis vectors.