Before physics discovered atoms and elementary particles (including photons) one could state that physics was the study of experiments and observations, i.e. measurements of variables with their errors, and modeling the observations mathematically. That is the way we got classical mechanics, classical electrodynamics and thermodynamics. The theories were formulated as second degree differential equations and the variables ,( position , momenta and energies for mechanics) were theoretically predictable and experimentall measurements only limited by the accuracy of the technology.
The solutions of second degree classical equations often have sines and cosines , and these described well waves in water and pressure waves and electromagnetic waves.
Then black body radiation, atomic spectra and the table of nuclear elements demonstrated experimentally the existence in the micro dimensions of structures that would not obey the classical theories. Energy appeared in packets ( spectra of atoms) and there were complicated rules of how protons and neutrons bound themselves into nuclei.
To start with the spectra in atoms were explained by the Bohr (planetary) model, a classical solution. In this model the electron around the proton forming the hydrogen atom and giving the Balmer (and other) series spectrum, a fit to the experimental observations, had to be constrained to fixed orbits by hand/postulate, and transitions from one orbit to the other released the observed light/photon spectrum. (by then the photoelectric effect had convinced that light was composed of photons).
This was unsatisfactory because even though stable orbits could be calculated around a charge, there was no classical reason why the electron becoming unstable would not fall on the proton and disappear. Hence the need for postulates fixing the orbits.
Then came Schrodinger's equation and solution for the hydrogen atom that reproduced the observed experimental series but also gave a general framework for describing data in the microcosm.
It is a partial differential equation and its solutions have sines and cosines and that is why they are called wavefunctions. It was necessary though to postulate several unorthodox ( for classical physics) operations and interpretations: the value of the observable variables could be predicted only by operating on the wavefunction with the appropriate differential operator and taking the integral over the phase space, the expectation value. For the position x,
for the momentum, the operator is more complicated:
Because the solutions of the Schrodinger equation were often in sinusoidal form, the wave pattern could appear and interference phenomena as with classical electrodynamics.
Thus the quantum mechanical theoretical framework made sense when psi was interpreted as a probability distribution, and this interpretation is known as the Born rule.
The probabilistic interpretation of quantum mechanics has been validated by innumerable experiments and observations . To develop an intuitive understanding can only come by studying and working on quantum dynamical systems.