Explain how there can be electric charge inside matter even if the matter shows no electric forces

  The electrical charge inside matter is hidden really well from the atoms it is made up of. The charge outside the atom is canceled out because the atom has an attraction between the positive and negative charges on the inside, which hold it together.

To detect the huge electrical fields residing inside matter you would have to venture down to the sub atomic level.  Electric field lines join positive and negative charges that make up the atom.  But just outside the atom the positive and negative particles sum to zero net charge.  This is what's meant by neutral matter.  In fact there are great electrical forces at work holding atoms together even when you can't see it on a macroscopic level.