Why does greater orbital overlap mean a stronger bond? According to valence bond theory, orbital overlap produces a bond. However, I don’t understand why having greater orbital overlap renders a bond stronger. It’s intuitive, I suppose, but I haven’t been able to find an actual explanation as to why that is. Are there quantum mechanical effects behind this?
An intuitive explanation is given in the comment above. Better orbital overlap generally indicates more constructive, favorable interaction between the two atoms.

Let's consider a homonuclear diatomic bond (e.g. $\ce{H2}$). The qualitative understanding is similar for two different atoms, but the math from quantum mechanics is a bit easier with a homonuclear bond.



So there's one bonding orbital ($\sigma$) and one anti bonding orbital ($\sigma^*$).

The energy of the orbitals will be:

$e_1 = \frac{H_{11} + H_{12}}{1 + S_{12}}$
$e_2 = \frac{H_{11} - H_{12}}{1 - S_{12}}$

You can put in some numbers and see what happens (e.g., as the overlap $S_{12}$ increases from 0 to 1, the bonding interaction becomes more and more energetically favorable).

N.B. There's a slight asymmetry - the anti bonding orbital is higher in energy than the bonding orbital is lower in energy.