Why don't polar and non-polar compounds dissolve each other?
Very simply, you explain the reason for this solubility rule by taking in consideration the energy requirements for the breaking of intermolecular forces between the molecules in the solute and the solvent.

Note: this is only a simplified explanation as it also depends on other factors such as change in entropy

Here is some background information on intermolecular forces. In non-polar substances, there are dispersion forces between each molecule. These dispersion forces are relatively weak and hence only require little energy to break them. In polar substances, there are dipole dipole and hydrogen bonding (depending on the substance) between each molecule. These forces are much stronger than dispersion forces and require more energy to break. 

Now lets consider the following cases:

Non-polar Solute and Solvent

For the solute to dissolve, the dispersion forces between the molecules in the solute and solvent need to break. This only requires very little energy. However when the solute dissolves into the solvent, they are able to be make dispersion forces with each other. The making of these forces releases very little energy. So simply put, very little energy is required to break the forces and very little energy released when making the forces. Hence overall everything balances out and the process occurs.

Non-polar Solute and Polar Solvent (and vice versa)

For the solute to dissolve into the solvent, both dispersion forces and dipole dipole forces are broken which require large amount of energy. However the molecules in the solute and solvent are only able to make dispersion forces with each other (as they aren't both polar). This only releases very little energy. Therefore overall, more energy is required than released and hence the process won't happen.

Polar Solute and Solvent

For the solute to dissolve into the solvent, dipole dipole forces are broken which require large amount of energy. However when they do dissolve, the molecules in the solute and solvent are able to form dipole dipole forces which releases large amount of energy. Therefore overall, everything balances out and the process occurs.