Frictional force doesn't depend on surface area, but why does this application work? I know friction doesn't depend on surface area and the professor has been demonstrating the same in all the previous lectures. But in this lecture he shows an application where the friction helps in balancing a large weight (T2) with a much smaller weight (T1).  
He further says increasing the angle the rope is in contact with the cylinder increases the frictional force and helps in balancing a much larger weight $T_2$. Doesn't this contradict the fact that friction doesn't depend on surface area? 
Hope I gave all the details so seeing the video is not required but here it is.

 A: Frictional force does not directly depend on surface area, but it does depend on the normal reaction force.
Consider two cubic bodies made of same material. The bigger body will have more weight, and higher friction will act on it. This is not a direct consequence of the fact that the bigger body has higher surface area.
Generally bodies with higher surface area have more weight, so the amount of frictional force acting on them is higher.
A: Winding the rope around the cylinder, you are not only adding a contact surface. You are also adding a "normal" force (a force pressing the rope to the cylinder).
p.s. frictional force non-dependence of the surface area is a gross simplification, but it does not generally relate to the rope-cylinder problem.
A: No, friction force still doesn't depend on contact area.
Here, true that friction does not depend on the surface area, but any part of the rope in contact with the surface is not just a single body, each part has its own term of friction force, which adds up to give a huge amount of friction force.
A: The "capstan formula" written on the board in your picture demonstrates clearly that the result does not depend on the contact area.
The result is independent of the radius of the cylinder. It only depends on the change of angle of the rope around the cylinder. 
For a fixed change of angle, the contact area is proportional to the radius, but the ratio of the tensions in the rope does not change.
A: No frictional force does depend on the surface area of contact (which itself is dependent on the normal reaction). 
Here is a para from Halliday Resnic Krane:

On the atomic scale even the most finely polished surface is far from flat. Figure 5-13,
  for example, shows an actual profile, highly magnified, of a steel surface that would be considered to be highly polished.One can readily believe that when two bodies are placed in contact, the actual microscopic area of contact is much less than the true area of the surface; in a particular case these areas can easily be in the ratio of $1:10^4$.
  The actual (microscopic) area of contact is proportional to the normal force, because the contact points deform plastically under the great stresses that develop at these points. Many contact points actually become “cold-welded” together. This phenomenon, surface adhesion, occurs because at the contact points the molecules on opposite sides of the surface are so close together that they exert strong intermolecular forces on each other.

