Why does dielectric constant of crystal fluctuates in case of Frenkel defect in a crystal?
The explanation provided in the text you quote is of the "hand waving" sort. Disrupting the crystal structure via Frenkel defects will have a rather complicated effect. That the effect is complicated is illustrated (in a rather rudimentary fashion) in the wikipedia, from which I derived the following figure:



Clearly the dislodged cations are shifted into closer proximity of other cations, but also of anions, and the orientational effects are not simple.

So the text you quote does not provide an entirely satisfactory explanation, I agree.



Now consider your line of argument:


  Now, if we think like that here, dipole moment is charge multiplied by
  distance, and distance has definitely reduced between a positive and
  negative ion here, so dipole moment should decrease and consequently
  dielectric constant should decrease but why is the opposite happening
  here.


The scenario you describe in the paragraph makes sense. You can write the dipole moment $p$ for two point charges of opposite polarity $q$ and separated by distance $d$ as

$$p = qd $$

Note that, as the wikipedia explains, the value of $p$ is independent of the choice of reference point, provided the overall charge of the system is zero. 

Obviously decreasing $d$ decreases $p$. 

The problem is that this does not describe the situation layed out in the text you quote: 


  The closeness of like charges tends to increase the dielectric constant of the crystal.


Consider what it means to increase the closeness of like charges. Presumably if the crystal remains constant in volume and overall charge, it requires that opposite charges move apart. 



But this is still a hand-waving explanation. You need to attempt some calculations with a model to get closer to a convincing explanation.