It looks like there are a variety of different blends used for the lint rollers. I searched some patents which give the different ingredients to make the different types of elastomers. They are below. It is a starting point but without more specifics that's all I could come up with.

Patent information:
patents_1
patents_2
patents_3

My best guess would be that it works by some type of static attraction between the lint roller and the lint.

Another possibility is the lint roller is made of a soluble adhesive which washes a layer away along with all the connected lint when in contact with water. I think this is less likely though as all the pictures I see online are of a very smooth roller.