Why is the orbital angular momentum of a pi electron along the axis of two atoms' molecule one?
Let the molecular axis be the $z$ axis, A and B the two atoms forming the bond. The $2p_z$ AO has atomic quantum number $m_l = 0$ and so has $L_z =0$. The MOs formed from
2pz AOs are therefore $\sigma$ MOs, bonding and antibonding. 
The $p_x$ AO is a linear combination of $m =+ 1$ and $m = -1$ AOs and has $|m| =1$.

$2p_x \equiv \frac{(2p_1 + 2p_{-1}) } {\sqrt 2}$

$2p_y \equiv \frac{(2p_1 - 2p_{-1}) } {i\sqrt 2}$

Therefore the MOs made from the $2p_x$ and $2p_y$ AOs have $|m| = 1$  and are $\pi$ MOs.