Why do bases feel slippery?
They don't, in general. This observation really only applies to strong alkalis (water soluble bases) present at specific concentrations.
Strong bases are able to react with the fatty acids and oils that naturally occur on the surface of your skin. The product of the reaction (which is known as saponification) is effectively a soap, which is why it feels slippery.
But this only applies to strong water soluble bases (i.e. alkalis), it is not a property that applies to every substance that happens to be a base, and the concentration also needs to be sufficiently high to enable the reaction to occur.