In writing or speaking about something technical, scientific, financial, etc, 'hand-waving' is what you do when you have no real knowledge or data. The Duolingo team, in the previous paragraph, said that 'in theory' their machine-learning translation system can learn how to categorise reports from learner's reports (of its accuracy) to improve its accuracy. They then say that they 'don't need to wave their hands about' (assert without data or knowledge) that this is true, because they have actual data.
'Hand-waving' is often used in e.g. science fiction stories, when the author wishes to employ an nonexistent technology to make the plot of the story work, (e.g. the 'flux capacitor' in Back to the Future), or to by-pass some restriction imposed by e.g. the laws of physics.
If I want you to lend me a million dollars for my tech startup, I might handwave about how I am going to find the engineers and software developers, or whether the idea is really possible.
Hand Waving Meaning
(idiomatic) Discussion or argumentation involving approximation,
vagueness, educated guessing, or the attempt to explain or excuse
Example: The sales pitch sounded good, but there was a lot of hand
waving about the price.
Hand waving (idiom) - Hamari Dictionaries
verb [ I or T ] to behave or talk as if a problem, question, etc. is
not important or worth serious attention, for example by ignoring it,
discussing it in a very general way, or discussing something else
I feel like the questions got handwaved off by the speaker.
His talk handwaved over some important details.
We handwave this inconsistency by assuming it was a revision error.