The word means to act like an uncle. Uncles might wish you well, might even go out of their way to help you, but they don’t feel the same responsibility or dread of shame regarding your behavior. They’re not your father, they’re not your boyfriend. They’re the brother of one of your parents.
If I can trick myself into feeling and acting in an avuncular capacity towards all the people I encounter who aren’t my nieces or nephews, then I will truly be a happy man. I will be giving without thought of reward.
I’ve met people much richer than I who when I say something like “You’re pretty well off” counter by insisting “Me, I’m just barely getting by. Now that guy over there, he’s well off!” Rich people who can’t control their spending don’t feel rich. Those who dread losing everything don’t sleep well at night.
Little avuncular gestures send a message to the uncle in question that he’s not in dire straits, that there’s nothing to be afraid of, he can afford mild generosity every now and then.