Handwavy, as defined on Wikitonary is
(of a demonstration, proof, or explanation) Missing important details or logical steps, perhaps instead appealing to common sense, tradition, intuition, or examples.
Handwavy actions can be contrasted with deliberately spelling out assumptions which have not been (or cannot be) proven, taking time to define terms that may not be mutually understood, and admitting where there are problems or difficulties when they are pointed out. While the Waver of Hands moves to make difficulties of their position vanish, the opponent of the Waver of Hands spells things out clearly such that their position is more easily falsifiable. An argument which invites criticism can stand up or fall to reason and evidence, while a handwavy position sneers from the sidelines of the forum without putting any skin in the game.
You have likely encountered handwaviness before and it was likely very distasteful to you.
Let us begin with an example of what is very often a form of self-deception: New Year's Resolutions. In late December, good old Old Self meets Handwavy Self and they have this conversation:
Old Self: We need to make some changes. I really wanna be healthier. What should I do?
Here, the Handwavy Self omits important details—for instance what should actually be done—and instead appeals to common sense and tradition. Handwavy Self did suggest eating healthier. But what does that mean? For eating healthier, a clearly spelt out measure such as cutting out soft drinks or added sugar, refraining from eating red meat 6/7 days of the week, or fasting once a week would suffice to turn a handwavy non-commitment into an actionable plan that Old Self has the opportunity to either succeed or fail at. But by failing to specify any terms for success or failure, the wily Handwavy Self creates a game where it is impossible to lose (or win, for that matter).
Handwavy Self suggests making a New Year's Resolution. But, if this is basically lip service—empty words to get re-elected—then what good does this do? Especially for the person that has a history of failing to keep New Year's Resolutions, making another New Year's Resolution is making a resolution to handwave. Rather than putting forth an effort to put a real plan together, Handwavy Self sells Old Self a snake oil non-solution, brushing aside concrete problems with vagueness and deferral of responsibility.
Look at the horizon at the beach. It's curved, isn't it? If you watch a ship, it can pass over the horizon. How do you explain that? Come on, everybody knows the earth is round.
This is a hand-wavy explanation for round earth.
For positions that we already agree with or otherwise have some commitment to want to be true, it is very easy to get intellectually lazy and give hand-wavy explanations. When skeptics push back, will you be able to defend your position without handwaving?
The handwavy way is to semi-engage, to kind of but not really have a conversation. It is to let good arguments go by unappreciated, and non-arguments be treated the same as well reasoned and supported positions (wow! such diversity of opinions we have here in the classroom today).
At its best, it's mildly annoying. At its worse, it's enabling if not encouraging very bad behavior.