The picture one in below is from the Jost's Riemannian geometry and geometric analysis .
$X$ is smooth tangent vector field. So , $X(c_p(t))$ is local Lipschitz. So, there is unique solution by Picard–Lindelöf theorem, but how to know the solution is smooth ?
Since $c' = X \circ c$ and both $X,c$ are differentiable, the chain rule tells us $c'$ is differentiable with $c''= DX \circ c'.$ But now we know $c$ is twice differentiable, so (assuming $X$ is too) we can differentiate again to see that $c$ is thrice differentiable. If $X$ is smooth you can repeat this as many times as you like, so $c$ is smooth.