A blotting paper absorbs ink by capillary action which is a natural phenomenon exhibited by liquids in the attempts to reduce surface tension.
Any thin tube having a very fine ne bore (say, less than one mm) is called a capillary tube. When such a tube is immersed n a liquid rises inside the tube to a level higher than that outside it. (Only liquids which have an angle of contact less than 90 degrees will show a rise. Other liquids like mercury will show a dip.)
Blotting paper is made of cellulose obtained from the pulp of cotton linter, wood or straw. A purified pulp paste is directly pressed to form blotting paper sheets without any treatment. So as to incorporate microscopic capillaries. When such a paper is placed in ink, water or any aqueous solution, the solution immediately enter in to these capillaries and spreads all over the sheet.
Capillary action is also responsible for many common phenomenon seen in our everyday life. It helps plants in raising the sap from the roots to the top through the stem. It also makes kerosene or oil in wick tip where it is burnt.