Why does water not leak from the bag? In the picture, the plastic bag is filled with water. The sharpened pencil penetrates through the bag. However, there is no leakage. 
Could you please explain why this happens?

 A: This experiment is described in detail on this site and many others.
The explanation focuses on the flexibility of the polymer molecules of the bag, sealing the gap around the pencil. 
To that I would add that, due to the surface tension, for the water to leak through a gap of a given size, a certain outward pressure is required - to overcome the inward pressure due to the surface tension. 
An estimate of such pressure for a round hole is presented in this post and it shows that, for the water to leak through a $0.1$mm round hole (about the thickness of printing paper), an outward pressure corresponding to a $10$cm water column is required. 
Assuming that this estimate is roughly applicable to the width of the gap around the pencil, we can easily see why this experiment works with common size plastic bags: there is virtually no gap to speak of and a more significant pressure would be required to stretch the plastic around the pencil in order to create one. 
In a specific example of the experiment shown on the picture in the post, the water hardly covers the holes, so the excessive outward pressure is close to zero: the easiest case.   
