Why does the blackbody radiate even at thermal equilibrium? Everytime, I read about blackbody, I always get confused at the point where it is written

Under thermal equilibrium conditions , the blackbody radiation depends only on temperature. ..... . At equilibrium , electromagnetic waves bounce around with the walls of the cavity.

What is this equilibrium all about? Why is it important here?? If there is thermal equilibrium, why does the body radiate?? Please help.
 A: The equilibrium mentioned in that quotation is between the radiation field and the walls of the container.  The walls of the container are imagined to be held at some fixed temperature by some (unspecified) means.   Under those conditions, the spectrum of the radiation in the cavity is that of a blackbody at the temperature of the walls.  The radiation field and the walls exchange energy, but since they are in equilibrium the temperature of both remains the same.  The walls radiate and absorb equally.
So far I haven't mentioned anything about radiation that leaves the cavity.  In the ideal case of a closed cavity, the discussion is about the walls of the cavity and the field within.  One could poke a small hole in the cavity, allowing some of the radiation to escape.  In the limit that the hole is infinitesimally small, the disturbance caused by the hole is negligible, and the radiation that leaks out has very nearly a blackbody spectrum.  This is a blackbody radiator.  Usually the temperature of the blackbody is much higher than the surroundings into which the radiation radiates.  Thus the blackbody radiator is not in thermal equilibrium with it's surroundings.
A: Any physical body has many degrees of freedom, not only mechanical, but also the field degrees of freedom. The energy is distributed amongst these degrees of freedom, so the radiation (as field excitations) is always present in a body. The energy exchange is always on and in the equilibrium conditions what is "radiated" is "absorbed" with the same rate. It is a dynamical equilibrium, like vaporization and condensation in a closed volume containing a liquid. On average there is no energy "flow" in a certain direction, but it is due to equality of energy fluxes in the opposite directions.
A: If a body has a positive temperature then it will emit radiation (over some very low limit perhaps). 
Imagine a hollow sphere at a uniform temperature $T$. radiation will be emitted from the walls inside the sphere and absorbed by the same walls. 
Consider a small part of this wall it loses energy by emission of radiation and gains energy by absorption of radiation from other parts of the wall. - Overall there is 'equilibrium' because all parts of the wall are at the same temperature and emit the same range of wavelengths with similar intensities. 

What is this equilibrium all about? 

as described above it is about exchange of energy between bodies of the same temperature.

Why is it important here?? 

Because if different parts of the system had different temperatures then they would emit different intensities/wavelengths of light - this would transfer energy from the hotter to the cooler body and mean that the spectrum of light was not 'blackbody radiation'

If there is thermal equilibrium, why does the body radiate?? 

All bodies will radiate photon unless they are really cold and very close to or at absolute zero temperature.
