What is the meaning of an overlapping of wave functions of two protons? A highschooler trying to understand what happens when 2 wavefunctions overlap in account of 2 protons being accelerated towards each other and smashing them together.
 A: 
what happens when 2 wavefunctions overlap in account of 2 protons being accelerated towards each other and smashing them together

It is called two protons scattering.
A wavefunction is a quantum  mathematical model . It is a complex value function, and to see what it describes one has to calculate the complex conjugate square of the function so as to get a real number function. This squared function, by axiomatic postulate, gives the probability of finding the particle at (x,y,z,t). 
When there are more than one particles to describe, a new calculation has to be made, to give a wavefunction, in this case , of two protons interacting . The calculation of what  happens is done with quantum field theory, and it will depend on the energy carried by the two protons.
If the energy is low, they scatter off each other according to the probability calculation, into different directions. Because they are charged, some energy might come off in terms of photons.
As their energy goes up, the probability of scattering goes up and with enough energy  of particle antiparticle pairs can be produced,  as studied in proton proton colliders, the last one being the LHC collider.. The studies extend the knowledge of particle physics .
