How do I understand the effect when the load current is over the inductor's saturation current?
Magnetic cores gradually saturate; some faster than others. In other words, their magnetic permeability lowers as the magnitude of the current increases: -



Picture source.

This effect is hardly noticeable at low current magnitudes (the blue curve) but as current and wound turns increase to form a bigger H field, the saturation is more clearly seen. There are also magnetic hysteresis losses that rise as current magnitude increases but these are different to the reduction in permeability (the slope of the BH curve).

If permeability (\$\mu\$) lowers then so does inductance: -



As to why the permeability reduces with a rising current is down to the magnetic domains in the magnetic material - there is only so much twisting and turning they can do. Once they have aligned as per the left hand picture below there's no more room for further alignment: -



Picture source.