Why does greater orbital overlap mean a stronger bond?
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.