How can a body be transparent? I need the theoretical explanation for the same. Has anybody succeeded in doing that practically?
It is possible - so far theoretically, and yes, there are groups which do their research in this topic.
The key factor is called Metamaterials. Briefly, metamaterials are materials which have effective permeability or permittivity lower then 1. In electromagnetic theory it means that refraction index is negative and group velocity is higher than speed of light (be aware of that we are not talking about velocity of information, we are not breaking the causality law).
How it can be used? You have an object which you would like to make invisible. You will put a layer of this metamaterials around it. If an electromagnetic wave (light, RF, whatever) enters, it will be shaped around the object and will leave the metamaterial layer on the other side. Since the group velocity is higher than the speed of light there will be no delay.
In practice, there are several issues which are not solved yet. We can do it just for very narrow band of frequency (ok, you are invisible for radar but not for human sight). So far we can make it just for low frequency (around 20 MHz). We need power supply for the metamaterials and some other issues.
There are a couple links showing invisibly cloaks and the science to it. They are about couple of pages but it shows that there are real invisibility cloaks out there.
I saw an article a while back that shows that the cloak works by bending waves around the object. They successfully did the experiment to an extent. It wasn't completely invisible but it only made the object semi-transparent. I'm not sure where that article is but here are two others that explain invisibility.
In practice, This concept is impossible, why, because when you are invisible, one can not see you then you can not see them either, this is the principe no1 of light. Or you can think that an object can become invisble but it is not for human. Cloaking metamaterial is one of the best choise up to now for developping this concept, you can find the works of Dr Smith of Padilla or Engheta about this field. The most difficulty problem is that, when we work in the range of light frequency about THz, the dimension of metamaterial period becomes very small cause it must be comparable to the light wavelength, and by this the fabrication of the material has alot of challenges.