The source to body voltage has an direct effect on the threshold of the MOSFET. The relation between the two is given by the Shichman-Hodges model as
So, as the body voltage increases (positive), the effective threshold voltage of the MOSFET decreases.Hence for the same gate to source voltage a greater inversion occurs in the channel.
To put it in simple terms,
The negative voltage you apply at the body actually pushes more minority carriers in the substrate towards the channel area which is relatively at a higher voltage because of the applied gate voltage.
Since the minority carriers in the substrate move towards the channel region, they recombine with the holes to result in more depletion layer charge across the channel.
Presence of the depletion region means more(fixed) charges in our would-be channel region. Now before the inversion channel is actually formed, the oxide AND the depletion layer act like two of capacitors in series - both layers having different dielectric properties, with the gate and the substrate acting as the plates sort of. Before Vgs is sufficiently large to form the inversion channel(i.e.,before reaching our Vt), the gate electrode has to exactly "mirror" the charge that is present in the substrate - the whole point of a CAPACITOR.
Hence, the more charge you have on the substrate, the more charge you need to put on the gate electrode - which implies an application of a greater Vgs. Hence the presence of the greater Vt.