When you linger somewhere, you stay there for longer than normal, or longer than you're supposed to. If you and your friends linger over lunch, then you're all done eating and you really should go, but you stay, because you're having a good time talking together.
Here, Romeo and Juliet have just caught sight of each other. They're strangers: they haven't met yet. They're staring at each other, thinking, "Whoa, that person is beautiful." It's rude to keep on staring at someone, but they don't care. They're lingering. It's a lingering stare. Here they are, lingering by the fish tank, enjoying the sight of each other.
Things, too, can linger. If you got a splinter yesterday and it still hurts today, that pain is lingering. It should go away, but it's sticking around. It's a lingering pain.
If you cooked spaghetti in your kitchen last week and you can still smell it today, that smell is lingering. It should fade away, but instead it's still here. It's a lingering smell. All kinds of feelings and smells can linger, good or bad!