They were laughing because of how terse and vague Smith was with giving out information. He simply says he's "John Smith from back east". The implication is that 'John Smith' isn't his real name. Which Strozzi found obvious but didn't push for any more details.
Both movies, as well as the Japanese film "Yojimbo" were based, somewhat loosely, on Dashiel Hammett's novel, "Red Harvest.". The opening credits actually state that the film is based on Yojimbo.
