It's obvious how close the Italian word espressivo is to the English "expressive," which is exactly what it means in a piece of music. The voices and instruments are supposed to perform that passage with lots of expression or feeling.
You may have seen movies where characters try out for a musical role. Some of them sing not quite in tune. Others get all the notes right, but are boring to listen to because no emotion or feeling comes through in their singing. They aren't singing espressivo! In such movies, the hero or heroine gets the role because the director listening to them is moved to tears or excitement by their espressivo singing.