What it means
Dry begging is angling for something without ever actually asking for it, dropping heavy hints and complaints until someone offers it up themselves. Think posting how broke you are right before payday, or sighing about how nobody ever buys you flowers. It's the passive-aggressive way to beg while keeping your hands clean.
Usage examples
"She keeps posting that she's starving and skint, classic dry begging for someone to send food."
"Just ask me if you want a lift, the dry begging with all the sighing is doing my head in."
"He's been dry begging on Instagram all week, posting his empty fridge and hoping someone sends him money."
"Stop dry begging and say what you want, mate. All that hint-dropping is jarring."
Where it comes from
The phrase spread online to name the hint-don't-ask move. Dry, because you do it without the straightforward request, begging all the same.