Leshon bnai Adam is a talmudic expression, favoured by Rabbi Ishmael to explain verbal repetitions in the Torah. Dibrah Torah kileshon bnai Adam —“the Torah spoke in leshon bnai Adam” — meant that sometimes God chose to phrase the Torah so that we could understand it, even if it meant repeating a word.
Lashon (leshon when part of a possessive phrase) means language. Bnai Adam, sons of Adam, means “people”. Even a woman can be called ben Adam, a son of Adam.
Maimonides used the phrase to explain biblical anthropomorphisms. Expressions such as “with an outstretched hand” do not mean that God actually has a body. The Torah was speaking in leshon bnai Adam so that we would understand the nature of the Exodus from Egypt, for example. Ibn Ezra on Genesis 1:26 explains the need for anthropomorphisms and leshon bnai Adam as the only way for people to understand such lofty matters. All the biblical metaphors are a recognition of our need to compare strange concepts to the familiar.
Today, leshon bnai Adam means “laymen’s terms”, another example of an ancient expression flowing freely in the modern language, its historical weight unfelt by most.