Molecules have IMAFs that determine the melting and boiling points. What about with ionic substances? What keeps the different compounds together? Do they experience IMAFs?
The problem is that you are trying to apply a paradigm that was developed for covalent compounds (the distinction between covalent bonds and intermolecular forces) to ionic compounds. They don't play by the same rules. In an ionic compound, all the ions are held together by electrostatic forces, which mean exactly the same thing as ionic bonds. There is no discrete molecular unit in an ionic compound. The word "molecule" does not apply.
Consequently, there is no such thing as an intermolecular force in an ionic compound. There are only ionic bonds.
I've heard that it's the ionic bond itself, but doesn't the bond remain after melting
Loosely speaking, it does not remain after melting. The reason why they have such high boiling points is therefore because of the fact that you have to break the ionic bonds for it to begin melting.
and even if it doesn't then how come covalent bonds don't cause the boiling and melting points in molecules?
Covalent bonds don't break during melting or boiling, with some exceptions (see: giant covalent compounds).