For the 1/r^2 law, it's because the flux of the electric field through a closed surface is constant. As r increases, the surface area of the sphere of radius r increases proportional to r^2 (specifically 4r^2). Therefore the flux per unit area, which is electric field, is inversely proportional to r^2.
Now for the dipole field. For a simple example, consider a dipole of two charges and points on the line that connects those charges. At a distance, the electric fields of the two charges are opposite and nearly equal in magnitude. They are not equal, because the distance to one of the charges is slightly further. Say the distance to one of the charges is r, and the distance to the other is r + dr. Then the total electric field is proportional to 1/(r + dr)^2 - 1/r^2. The limit of that as dr -> 0 is the definition of the derivative of 1/r^2, which is -2/r^3. So the electric field from the dipole is proportional to 1/r^3.