I came upon this question as I was going through the concepts of tension. Well according to Newton's third law- every action has an equal and opposite reaction. Here my question is that if the tension at point B balances the tension at point A then which force balances mg as it can't be balanced by the reaction force of mg which attracts the earth towards the mass as it is not in contact with the mass. Then why doesn't the mass fall?
It is the normal force ( https://en.wikipedia.org/wiki/Normal_force ) of the Earth against the supports that hold the platform up. Normal force is what keeps our feet from sinking into the Earth, due to our weight, it keeps a book on a table from falling through the table. It will keep the supports from falling into the Earth, the supports hold the platform, which holds the string and mass.
There are two forces acting on the mass are gravity and tension from the rope. This is seen in your free body diagram on the right side of your image. The mass does not fall (or rise) because these forces are of equal magnitude and opposite direction, thus the net force acting on the mass is $0$.
Yes, if we look at the rope the force pulling it down at $A$ is equal and opposite to the force pulling it up at $B$, but this doesn't make the force at $A$ suddenly nonexistent for the mass. There is a downward force acting on the rope at $A$, and by Newton's third law this means there is an upward force acting on the mass at $A$. This is the upward force discussed above. Note that the forces at $A$ and $B$ acting on the rope do not form a Newton's third law pair.