Well, what is the meaning of wave function? What does it represent? In Schrodinger's equation, we find the value of Ψ. But what is Ψ exactly? Max Born only gave an explanation of what $Ψ^2$ (the probability of particle with low mass enough to have an observable wavelength to be in a position in space). But no one knows what Ψ is.
The name for what the wave function represents is “probability amplitude”. You should simply consider that a fundamental physical quantity (at least in non-relativistic QM), no more or less mysterious than “electric field”. Electric field is three real numbers at each point in space. Probability amplitude (for one particle) is one complex number at each point in space. Why? Because this is how Nature seems to work. Do we really “know” what electric field or probability amplitude is? No. We just learn how they work.
Physics builds mathematical models of reality. What reality is is probably unknowable or meaningless, and more philosophy than physics.