A base emulsifies fats and oils. Your hands are covered in oils that your skin produces naturally. A base on your skin will dissolve those oils producing the slippery effect.
Bases turn the esters (the bonds betwen glycerol and fatty acids in fats) into an alcohol and the salt of a carboxylic acid. In this case the alcohol is formed from the fatty acid. This means it has a long hydrocarbon chain, which is lipid soluble, and an OH group which is water soluble. This polarity is the key property of soap. Hence getting bases on your fingers, turns your fingers to soap and becomes slippery.