The phrase "πιάνει δροσιά" means "αρχίζει να έχει δροσιά" ("it becomes cool").
In the same way we say "πιάνει ζέστη" = it becomes hot, "πιάνει κρύο" = "it becomes cold", "πιάνει βροχή" = "It starts raining".
Other expressions with the verb "πιάνω":
"με πιάνουν τα γέλια" = "I start laughing"
"με πιάνουν τα κλάματα" = "Ι start crying"
"με πιάνει απελπισία" = "I become desperate"
"με πιάνει το παράπονο" = "I become sad"
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Vasiliki Baskou, Instructor/Director, https://learn-greek-online.com.