The point of a convnet, or many kinds of neural networks in general, is to go from a lot of data down to a small piece of data. In classification tasks, for example, the input is all the pixels that make up a picture of a house, and the output is just the word "house" (or rather a number representing the word "house").
Obviously this process loses information. If you go from the word "house", back to a picture (i.e. you tell it "draw a house") you're probably going to get a completely different house!
In the style transfer task we have many numbers to describe the house picture with, not just the word "house", but we still have less than the full pixel data. Imagine that the intermediate representation represents something like "yellow wooden house with three windows and one window above and a red brick basement with 4 windows and the house is drawn at a 30 degree angle and there's a pink house to the left with two small windows below and one window above and a red roof is visible behind and above the pink house and ....."
If you had that representation, you could try to draw the original picture again, and the more information you have, the more accurately you can draw it. Early layers of the convnet contain information like "there's a vertical line at pixel coordinates 123,456" and later layers contain information like "there's a yellow house at pixel coordinates 123,456", and if you get to dense layers, they may just say "there's a yellow house".