I read this expression in a text. It seems to be an old proverb about monkeys. But I am not sure about its meaning. Why 筋, for instance? Does it mean muscle? reasoning faculty? Why would they be lacking?
It means: "Monkeys are less clever than humans" or literally: "Monkeys have 3 hair less than people"
足らぬ being used for 足りない
and 筋 being the counter for hair
To add a bit of history/legend:
毛 doesn't stand for "hair" but for apparel, appearance or skill...etc.
The legend states that during the creation of the world, Kamis (god(s)) thought human and monkeys were too similar and removed 3 details (毛) from them.
A monkey asked to "god" (kami) why they could not become human. God answered, because you lack the following 3 points (毛):
(reason, compassion and sense of accomplishment)
猿は人間に毛が三筋足りぬ、is also said 毛が三本足りぬ. I think 三本足りぬ is more popular than 三筋足りぬ as a saying today.
Both '三筋' and '三本' mean three pieces (or threads) of hair. It means monkeys resemble men, but they have three pieces short of hair as compared with men, meaning monkeys are inferior to men.
It is an interesting concept and simile that not the size of brain or the level of intelligence but the number of hair makes a decisive distinction between men and monkeys.
When we say 'あいつは毛が三本足らぬ,' it means he is stupid like a monkey. We used to say this for abusing our boss and colleagues over drink after five. Sorry for a clever monkey.