Yup, that word is indeed etymologically related to babili. It's an old word that appears in most (or all?) Germanic languages, including English (babble), German (babbeln), Dutch (babbelen), Swedish (babbla), etc, etc.
But as the i in the middle gives away, the Esperanto word specifically comes from the French word babiller, which in turn likely entered the French language through a dialect of Old Dutch, during or before the middle ages.
The question was used to clarify the meaning of the English translation, Let us chat, which has two distinct meanings in English:
Allow us to chat (Let us chat)
first-person plural command (Let's chat/Let us chat)
It comes down to whether the sentence is directed at the person you want to chat with or at a third party who is interrupting.