Consider a Riemannian manifold $(M,g)$ with the Levi-Civita connection $\nabla$. If $D_t$ is the covariant derivative along curves descending from $\nabla$, a geodesic is a curve $\gamma: I\subseteq\mathbb R\longrightarrow M$ such that $D_t\gamma'=0$ where $\gamma'$ is the velocity vector field along $\gamma$. In coordinates (respect the coordinare frame $\frac{\partial}{\partial x^1},\ldots\frac{\partial}{\partial x^n}$) a geodesic is a curve that satisfies the following equation(s):
$$\ddot \gamma^k(t)+ \dot\gamma^j(t)\dot\gamma^i(t)\Gamma^k_{ij}(\gamma(t))=0$$
It is evident that there aren't constraints for the parameter $t$, but on the book "Carroll - Spacetime and relativity" I read the following mysteriuous phrases:
Notation: Here by the parametrization with the "proper time" $\tau$, he means the parametrization with the arclength parameter. The equation $3.44$ is the same that I've just written above and moreover the others cited equations deal with the variational approach to geodesics.
So I don't understand why, according to Carroll, if a curve satisfies the above equation(s), then its parametrization should be affine. The author doesn't explain this point.
To elaborate on the comment by @PavelC, the geodesic equation $\ddot \gamma^k(t)+ \dot\gamma^j(t)\dot\gamma^i(t)\Gamma^k_{ij}(\gamma(t))=0$ actually implies that the curve is constant speed. This is what the author seems to be saying: the parameter is necessarily affine with respect to the unit speed parametrisation. The proof of this can be found in my course notes here, Lemma 8.3.5 on page 67.
On an arbitrary surface $M$, a curve $\beta$ satisfying the geodesic equation is necessarily constant speed. It suffices to prove that the square of the speed has vanishing derivative: $$ \begin{aligned} \frac{d}{ds}\left(|\beta'|^2\right)&=2\langle\beta'',\beta'\rangle \\&= \langle L_{ij} {\alpha^i}' {\alpha^j}' n \,, {\alpha^k}' x_k \rangle \\&=L_{ij} {\alpha^i}' {\alpha^j}' {\alpha^k}' \langle n, x_k \rangle \\&= 0. \end{aligned} $$