Full of tree sap and moisture, or (more commonly) overly emotional, nostalgic, or silly in a corny way.
From 'sap' (the liquid inside plants) + '-y' (adjective suffix). Originally meant 'containing sap,' but by the early 20th century it evolved to mean 'emotionally weak or overly sentimental,' perhaps because sap is sticky and weak.
When people call a romantic movie 'sappy,' they're basically saying it's so full of gooey emotion that it's uncomfortable—and interestingly, real maple tree sap becomes gooey and sticky as you heat it, so the word's double meaning actually makes sense.
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