Hand-kissing is a very old and traditional way of showing respect and admiration towards someone else. It is done by gently taking someone's hand and bringing it up to your mouth to give it a small kiss. It is kind of like saying "hello" in a special and polite way.
Think of it like when you greet your grandma or grandpa and give them a big hug or a kiss on the cheek when you see them. Hand-kissing is like that, but more fancy and grown-up. It's like saying "I think you are really special and important, and I want to show you that by being extra respectful."
People often use hand-kissing when meeting important people like royalty, politicians, or celebrities. It is also sometimes used in romantic situations, like when a man wants to show his love and affection for a woman.
Hand-kissing is not as common today as it used to be, but it is still used in some formal situations or by people who appreciate the old-fashioned manners and customs.