Riemann surface with transition functions of the form $z \mapsto az+b$
This is Lemma 1 in
Gunning, R. C., Special coordinate coverings of Riemann surfaces, Math. Ann. 170, 67-86 (1967). ZBL0144.33501.
Gunning proves this by observing that the canonical bundle of the Riemann surface $\Sigma$ has to have zero 1st Chern class (since the transition functions of the canonical line bundle are constant), hence, $deg(K_\Sigma)=0$. Since for a general compact connected Riemann surface $X$, $deg(K_X)=2g-2$, in our case the genus of $\Sigma$ has to be 1.
If you are interested in this staff, I suggest reading the entire Gunning's paper.