The phrase likely originates from the idea of a very small violin producing a high-pitched, squeaky sound that may sound comical or exaggerated in comparison to the seriousness of the situation being described. It suggests that the speaker believes the person's reaction or expression of distress is overblown or not deserving of genuine sympathy.
When someone says they are "playing the smallest violin" for another person, they are essentially saying that they find the person's complaint or misfortune unimportant or even amusing. It is a way of dismissing someone's concerns or problems without showing any real compassion or understanding.
It's important to note that using this expression can be perceived as insensitive or dismissive, so it should be used cautiously and only in appropriate contexts where the intent is clearly understood.