Say, I have a ladder which is kept slanted across a wall. It is obvious that it the ladder were to fall, it would rotate about the point which is in contact with the ground. How do we justify this?
In a freely falling body, the rotation is always about the center of mass right? Then why should it fall about that particular point?
The ladder rotates about the point of contact with the ground because, given sufficient friction, that point does not move relative to the ground. The other points of the ladder do move. Since the ladder can assumed to be rigid we get rotation about the point of contact with the ground.
How do we know the point about which a body experiences torque?
We don't know. Or to put it a better way, there is nothing to know. The point about which we calculate the torque of a force about is completely subjective. There are points that are better to pick than others depending on what you want to know about the system, but there is no law, rule, etc. that says you have to calculate the torque of a force about some specific point.
In a freely falling body, the rotation is always about the center of mass right?
I'm assuming you mean freely floating, perhaps? Or I guess as long the the gravitational field is uniform then that's fine too. But I will assume there are no external forces acting on our rotating body. Then you are right, it must rotate about it's center of mass. This is because if the object rotated about any other point, then it's center of mass would be accelerating. But this violates Newton's laws, since the center of mass of a system of particles cannot accelerate if there is no net external force acting on that system. Therefore we must only have rotation about the center of mass if no net force acts on the system.
Your ladder has the external forces of gravity, the floor, an friction acting on it. Therefore, the ladder does not rotate about it's center of mass.
The need to choose a reference point around which to measure torque is the same as the need to do so when measuring angular momentum. In fact, torque is the time-rate of angular momentum, just as force is the time-rate of linear momentum.
The intuitive core is that, in effect, when you consider an extended, rigid body, what can be seen as purely rotational motion around one point can be considered as some combination of rotational and translational motions around another.
Any point will, thus, suffice to calculate both torque and angular momentum - it just may be that, depending on the choice thereof, they will then behave in a "weird" way because psychologically, you've already identified a more "natural" point around which to compute them. But them behaving weirdly in this case doesn't mean you did the maths wrong; just that you found a less-than-ideal way to describe the situation.
