In the minimal interpretation, i.e., the one you need to apply the quantum theoretical formalism to observables and the only piece of interpretation which is relevant as far as physics is concerned, the "wave function" has the meaning and only the meaning of a probability amplitude. Let's concentrate on the single-particle case for simplicity first. You have a function ##\psi(t,\vec{x})## which fulfills and equation of motion, the Schrödinger equation, and ##P(t,\vec{x})=|\psi(t,\vec{x})|^2## is the probability distribution for finding the particle at position ##\vec{x}## when measured at time ##t## (Born's Rule).
What does this mean in practice. First, to solve the Schrödinger equation you need the initial state, represented by ##\psi(t=0,\vec{x})=\psi_0(\vec{x})##. This you get by preparing the particle in a specific way. For a double-slit experiment you prepare the particle with a quite well defined momentum, i.e., a wave packet that's well-peaked in momentum space, while in position space it's pretty broad (the position-space wave function is the Fourier transform of the momentum-space wave function). Now you put a double-slit in the way of this particle and in some distance from this double slit you put a screen which registers the position where it is hit by the particles. A single particle makes a single dot on the screen, not a broad distribution according to ##P(t,\vec{x})##. This underlines that Born's Rule is a (and as far as I know so far the only) definition of the physical meaning of the wave function consistent with observations.
Now to verify the interference pattern, predicted by the Schrödinger equation for the double-slit experiment, you have to prepare many single particles in the same way as described above. Then you can measure the distribution of the particles running through the double slit and compare it with the predictions of the formalism. Quantum theory turned out to be right. That's it from the point of view of physics.
Now you can bring up various questions about this, particularly the question what a microscopic particle (perhaps even an elementary particle like an electron) "really" is. Quantum theory tells you, it's neither a little lump of matter behaving like a classical "particle" in the sense of Newtonian (or relativstic) classical mechanics, i.e., as a determined trajectory in phase space. The interference pattern predicted by the Schrödinger equation and verified in experiment tells you that indeed the particle, if prepared in the way described above, is really not localized in the classical sense, and since the wave function has an overlap with both slits in the double slit such that you cannot predict accurately that it must come through the one or the other slit (and you also make no other attempt to mark through which slit the particle goes), you get this interference effect. On the other hand, as stressed above, a single particle always appears as a single point i.e., it interacts with the screen in a small region but not as a washed-out continuum of intensity (of whatever) as in a classical field theory like classical electrodynamics (for the em. field what's measured as intensity is the energy density of the electromagnetic field). Thus it doesn't make sense to say that the electron is something described by a classical Schrödinger wave whose square is a kind of intensity (in this case its square would be something like the particle-number density in a fluid, but that's a contradiction in itself, because the single-particle wave function is supposed to describe a single particle not (very) many as the particl-number density in fluid dynamics does).
So Born's interpretation as a probability distribution for the position of the particle is one that is in accordance with all observations and, in my opinion, also so far the only convincing one among all the various "interpretations", i.e., metaphysical additions one uses on top of the physical minimal interpretation to satisfy some philosophers of science in their seek of "an ontology" of the microscopic world, which behaves according to the laws of quantum physics rather than classical physics. From the point of view of a physicists who cares about physics only, that's a pretty hopeless endeavor, because the different variations of interpretations (there are countless more or less esoteric flavors of interpretations like Copenhagen interpretations, Bohmian mechanics, many-world interperations,...) cannot be tested by objective experiments since they all predict the same outcome as the minimal interpretation, and that's why I consider the minimal interperation the only one needed by physicists.
