Water has intermolecular forces of Hydrogen bonding one of the strongest IMFs and thus has a high viscosity and surface tension compared to other liquids. The adhesive forces between water over take the cohesive forces between the paper and water and allow the water to move up the paper and the ink as well. A common everyday example is paper towels. The way a drop of water spreads through the paper towel is capillary action.
Additional info on capillary action: