Intuition for the overlap integral
Let $\Phi, \Psi$ be normalized wave functions.

Then $0 \le|\langle\Phi\vert\Psi\rangle| \le 1$, with $\langle\Phi\vert\Phi\rangle = 1$ and $\langle\Phi\vert\Psi\rangle = 0$ for orthogonal $\Phi$ and $\Psi$.

Since we can argue that any wave function completely contains itself and for two orthogonal wave functions none of them contains a part of the other, we can interpret the scalar product $\langle\Phi\vert\Psi\rangle$ as a measure for overlap.