A body is attached to another with a string and both are in free fall. How does the tension in the string being zero make sense conceptually?
@Bill N has shown you the analytical approach. Here is a conceptual approach that might help.
Think about a tug of war between two teams pulling on a rope. There is tension in the rope and the distance between the teams is constant. Now imagine you are in the middle of the rope and cut it. The teams will fly apart which confirms there was tension in the rope that kept them from separating.
Now imagine two objects attached by a string one vertically above the other with the string stretched out in free fall. The distance between the objects is constant. Now imagine you are also in free fall between the two objects. You cut the string. Will the distance between the two objects increase like in the tug of war? No, the separation will remain the same because the acceleration of each object, $g$, will remain the same. Conclusion: there was no tension in the rope.
Hope this helps.
If the tension were not zero then the objects would not be in free fall. Free fall means that there are no non-gravitational forces acting, and tension is a non-gravitational force.