When a body of density d is immersed in a fluid of density p, then it sinks. i.e. d>p.Then why does the level of liquid falls???

The level of liquid does NOT fall. Instead it increases, you can consider any day today example to prove this.

Archemedes just said that:

When an object is immersed in a fluid, it is buoyed up by a force equal to the weight of the fluid displaced by the object. This became known as Archimede's principle. The weight of the displaced fluid can be found mathematically. The fluid displaced has a weight W = mg. The mass can now be expressed in terms of the density and its volume, m = pV. Hence, W = pVg.

So this displaced fluid will always increase the level of fluid.

Approved

In science, buoyancy (pron.: /'b??.?nsi/) is an upward force exerted by a fluid, that opposes the weight of an immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus a column of fluid, or an object submerged in the fluid, experiences greater pressure at the bottom of the column than at the top. This difference in pressure results in a net force that tends to accelerate an object upwards. The magnitude of that force is proportional to the difference in the pressure between the top and the bottom of the column, and (as explained by Archimedes'' principle) is also equivalent to the weight of the fluid that would otherwise occupy the column, i.e. the displaced fluid. For this reason, an object whose density is greater than that of the fluid in which it is submerged tends to sink. If the object is either less dense than the liquid or is shaped appropriately (as in a boat), the force can keep the object afloat. This can occur only in a reference frame which either has a gravitational field or is accelerating due to a force other than gravity defining a "downward" direction (that is, a non-inertial reference frame). In a situation of fluid statics, the net upward buoyancy force is equal to the magnitude of the weight of fluid displaced by the body.^[1]

The center of buoyancy of an object is the centroid of the displaced volume of fluid.