I don't hear this phrase used often anymore; of course it was common many years ago when people still actually sent postcards instead of Instagram posts. Since it's not used often now, trying to find the origin and meaning behind it has resulted in conflicting or vague descriptions.
So what does it mean, and does it have more than one meaning? For example, it might be genuine request for answers to a proposed question, but they're expected to be brief. However, I've heard it being used in a sarcastic tone, like "Will Covid-19 cause a recession? Answers on a postcard, please."