What is the physical meaning of multiplication of two wavefunctions?

It doesn't really mean anything physically (as far as I know.) Rather, it's a mathematical trick that allows us to explicitly figure out how to write any wavefunction $psi$ as a superposition of some set of eigenstates $phi_n$.

Specifically, the property of completeness says that any wavefunction $psi$ can be written as some superposition of the eigenstates: $$ psi = sum_j a_j phi_j $$ for some set of complex coefficients $a_j$. The orthogonality property is then $$ int phi_i^* phi_j , dx = delta_{ij} $$ (note the complex conjugate, though if the wavefunctions are real you can ignore it.)

These two properties allow us to extract the coefficients $a_i$ for an arbitrary wavefunction $psi$, as follows: begin{align*} int phi_i^* psi , dx &=int phi_i^* left[ sum_j a_j phi_j right] , dx &= sum_j a_j left[ int phi_i^* phi_j , dx right] &= sum_j a_j delta_{ij} &= a_i. end{align*} So if you tell me that the wavefunction at some time is $psi$, then I can figure out exactly what superposition of the eigenfunctions it is by doing a bunch of integrals. Knowing the coefficients $a_i$ then allows me to figure out the probabilities and expectation values of measurements, or (if the eigenstates are energy eigenstates) to use the time-independent Schrödinger equation to find how the state $psi$ evolves in time.