The selection rules for hydrogen are: $\Delta l=\pm1$ and $\Delta m=0,\pm1$. The first makes intuitive sense because of the conservation of angular momentum and the fact that a photon has spin 1. But does there exist an intuitive explanation for the $m$ selection rule as well?
The selection rule for $m$ corresponds to the conservation of the $J_z$ component of the angular momentum, where $z$ is the chosen quantization axis.
The $\Delta m=0$ possibility occurs when the system is driven by light which is linearly polarized along the quantization axis, in which case each photon has total angular momentum $1$ but nevertheless has $J_z=0$.
The $\Delta m=\pm1$ case happens when the system is driven by light which is circularly polarized in the plane orthogonal to the quantization axis, in which case each photon has angular momentum component $J_z=\pm 1$ that must be absorbed by the system.