Here are two arguments.

1. Suppose that $M$ is a noncompact complete connected Riemannian $n$-manifold with finitely generated fundamental group and Ricci curvature $\ge 0$. By the Bishop-Gromov inequality, volumes of metric balls in $M$ satisfy $$ Vol B(x,R)\le \omega_n R^n, $$ where $\omega_n$ is the volume of the unit ball in $\mathbb R^n$. Thus, $M$ has (sub)polynomial volume growth. From this, one concludes that the fundamental group $\Gamma$ of $M$ also has (sub)polynomial growth: $$ \# \{\gamma\in \Gamma: |\gamma|\le R\}\le C R^n $$ for some constant $C$. Here $|\gamma|$ is the word-length of $\gamma$ with respect to a fixed finite generating set of $\Gamma$. Now, by Gromov's polynomial growth theorem it follows that $\Gamma$ is virtually nilpotent (i.e. contains a nilpotent subgroup of finite index). If $M$ is 2-dimensional, it follows that $\pi_1(M)$ is abelian (actually, cyclic) and, thus, $M$ is diffeomorphic to the plane or to the annulus or to the Moebius band.

2. Let us prove that if $M$ is a complete connected noncompact 2-dimensional Riemannian manifold of curvature $\ge 0$, then either $M$ is simply connected or the metric on $M$ is flat. In the latter case, $M$ is isometric to a cylinder or to a flat Moebius band.

Indeed, assume that $M$ is not simply connected. Take a nontrivial element $\gamma\in \pi_1(M)$; then $\gamma$ has infinite order (since $M$ is assumed to be noncompact, which implies that its fundamental group is free). Take a Riemannian covering $p: N\to M$ such that $p_*(\pi_1(N))=\langle \gamma\rangle < \pi_1(M)$. It suffices to prove that $N$ is flat. Without loss of generality, we may assume that $N$ is orientable (otherwise, take its orientation covering space). Thus, $N$ is diffeomorphic to $S^1\times \mathbb R$. Now, take two sequences $x_n, y_n\in N$ diverging to different ends of $N$. Specifically, we can take $x_n=(1, n), y_n=(1, -n)$, $n\in \mathbb N$. Let $x_ny_n\subset N$ denote a minimizing geodesic connecting $x_n$ to $y_n$; I will parameterize this geodesic with unit speed, $c_n: [a_n, b_n]\to x_n y_n$, $a_n< 0< b_n$, $c_n(0)\in S^1\times \{0\}$. Clearly, $\lim a_n=-\infty, \lim b_n=\infty$. Then, after passing to a subsequence, the sequence of geodesics $c_n$ converges to a complete minimizing geodesic $c: \mathbb R\to N$. Next, I will apply the Cheeger-Gromoll splitting theorem: If a complete connected Riemannian manifold of Ricci curvature $\ge 0$ contains a line (a complete minimizing geodesic), then it splits isometrically as a product of $\mathbb R$ and another Riemannian manifold. In our case, this means that $N$ is isometric to the product of a circle and the real line, i.e. is flat. Thus, the original manifold $M$ was flat as well.

Now, it follows from the classification of complete flat 2-dimensional Riemannian manifolds that every such manifold $M$ is either a cyclinder or a Moebius band, or the plane.

I am sure, there are other arguments as well, these were the first two which came to mind. The result itself is probably due to Cohn-Vossen:

Cohn-Vossen, S., Kürzeste Wege und Totalkrümmung auf Flächen., Compositio math. 2, 69-133 (1935). ZBL61.0789.01.

Remark. Milnor conjectured that a complete Riemannian manifold of Ricci curvature $\ge 0$ has finitely generated fundamental group. This was recently disproven in

Bruè, Elia; Naber, Aaron; Semola, Daniele, "Fundamental Groups and the Milnor Conjecture". arXiv:2303.15347 (2023)