Why does transition state loses its ability to vibrate? (Transition State Theory)
Only some of the statements you quote are true, certainly a transition state cannot be isolated  since it lasts for less than a picosecond. In fact there is hardly any direct measurement of transition states, such as a spectrum or time profile or any structure. It is not that they are too fleeting to measure but that they occur very infrequently in any reaction so their population is always vanishingly small. This is a real experimental challenge.

What we do know is that the transition state is just like any other molecule with bonds vibrating/rotating  normally except for the bond(s) that are involved in reaction. They stretch and break then form new ones as the transition state is crossed. This pathway is the lowest point on the trajectory from reactants to products and is usually drawn at the top of the reaction energy profile as a 'saddle point'. (When at the saddle point looking at right angles to the reaction path the energy increases, looking forwards or backwards along the reaction path the energy decreases).  As the new bond has lots of energy it is initially vibrationally hot and rapidly loses energy either to solvent or into other vibrational modes in the product molecule or both. 

In a complex molecule you can imagine the reaction starting as the species approach one another. The energy starts to increase as the atoms that are going to react  approach one another, but collisions with solvent can randomly knock them off course, or further on course for a while, or vibrations in the molecule may randomly add or remove some energy and so hinder or accelerate the reaction. One can then imagine the species  behaving somewhat as if they are diffusing along the reaction path, sometimes going forwards but sometimes being hindered. At any point, including the saddle point, they can diffuse around a bit and either return to reactants or proceed to products.  

Even in simple atom-diatom reactions in the gas phase, such as $\ce{H + Cl_2}$ vibrations in the diatomic molecule  can inhibit reaction even when there is enough total energy to react; sometimes the energy is just in the wrong place and cannot aid the reaction. This can be the case because the diatomic is both rotating and vibrating and reaction will depend on the angle which it has to the line of approach ( atom to diatom) at the separation at which reaction starts.