Consider free NLS: $i\partial_tu+\Delta u=0, \quad u(0, x)=u_0$
The solution of this IVP, can be written as
$$u(x,t)=e^{it\Delta}u_0(x)$$
It is clear to me that how to prove following estimate:
$$ \|e^{it\Delta} f \|_{L^{p'}} \leq |t|^{-\frac{d}{2} \left( \frac{1}{p}-\frac{1}{p'} \right)} \|f\|_{L^{p}} \quad (1\leq p \leq 2)$$
My question is: Why this estimate is known as dispersive estimates?
To complete Piero's answer, let me give a definition of dispersiveness. To stay at a rather simple level, let me consider an linear evolution PDE with constant coefficients, $$P(\partial_t,\nabla_x)u=0.$$ Hereabove, $P$ is a polynomial. Let me assume a property of homogeneity, $$P(\lambda^\alpha\tau,\lambda\xi)\equiv\lambda^m P\tau,\xi).$$ It is met by the wave equation and the Schrödinger operator.
When studying the wave propagation, one faces the algebraic equation $$P(\tau,\xi)=0,\qquad\xi\in{\mathbb R}^n,\quad\tau\in{\mathbb C}.$$ This defines roots $\tau_1(\xi),\ldots,\tau_k(\xi)$ - when $\tau_j(\xi)$ is real, $\nabla_\xi\tau_j$ is a group velocity. In the best situation, they have constant multiplicities for $\xi\ne0$, and then are smooth functions. Then the equation is dispersive if none of these functions is linear, meaning that the velocities $\nabla\tau_j$ do vary with $\xi$.
Because $\tau(\lambda\xi)=\lambda^\alpha\tau(\xi)$ for $\lambda>0$, every equation whose order in time differs from the order in space ($\alpha\ne1$) is dispersive - mind however that if $\alpha<1$, then the Cauchy problem is not even well-posed. But even if $\alpha=1$, the equation can be dispersive, as shown by $\Box u=0$. Other examples come from linearized gas dynamics: the acoustic waves satisfy $\tau(\xi)=v\cdot\xi\pm c|\xi|$ ($v$ the fluid velocity, $c$ the sound speed) and thus are dispersive, though the entropy waves satisfy $\tau(\xi)=v\cdot\xi$, hence are not dispersive.
The fact that dispersive equations satisfy $L^p-L^q$ decay estimates is an extension of so-called Strichartz estimates.
The intutitive picture is the following: there is a certain amount of "mass", which at time $t=0$ you might imagine concentrated in a heap near a location $x=0$. This heap evolves with time according to a differential equation.
If the equation is a transport equation or a 1D wave equation (or the solution is a soliton for some suitable nonlinear equation), the heap moves in some direction but remains concentrated near some point $x(t)$. This behaviour is non dispersive.
But if the equation is of dispersive type, the heap also spreads as time evolves: the total mass will remain the same, by conservation of mass, but it will spread on a larger and larger region of space. This is a dispersive behaviour.
In your example, if you take $p=2$ you have conservation of mass, measured in $L^2$ norm. However, if you take $p=\infty$, you see that the "peak" of the heap becomes lower, at a rate of $t^{-n/2}$.
If you take as initial data a wave packet, you can actually see this effect: the packet moves at a speed proportional to the frequency (as postulated by quantum mechanics), it spreads (this is called decoherence in physics), and its height decreases precisely at the stated rate.
The word "dispersion" means "spread". In the context of wave equations it refers to the spreading (broadening) of a wave packet. The estimate in the OP is called dispersive because it bounds the spreading of the wave packet (left-hand-side of the equation) as a function of time (right-hand-side).
An instructive discussion of the use of "dispersion" and "dispersive" in connection with the non-linear Schrödinger equation is given in these lecture notes, page 2.
Schrödinger equations are classified as dispersive partial differential equations and the justification for this name comes from the fact that if no boundary conditions are imposed their solutions tend to be waves which spread out spatially. But what does this mean mathematically?
Another use of the word is "dispersion relation", which refers to the relation between frequency and wave number. A linear relation implies that the wave packet will not spread. A material with a nonlinear relation is called a "dispersive medium".
