"lean and mean" means to be ready and able to do something efficiently. Please see this link: http://idioms.thefreedictionary.com/lean+and+mean
The person's shirt claims (true or not) that the wearer is a person who is ready and able to drink great quantities of alcohol. In other words, the legend on the shirt metaphorically describes its wearer as a "machine" that is able to consume alcohol.
The "humor" comes from the rhyme: lean, mean, .... machine.