When you do or say something imploringly, you're trying to plead with someone for something. People act imploringly when they really need help.
Have you ever had someone look at you with big puppy-dog eyes, hands pressed together, saying "Please, please, help me with this?" They looked at you — and spoke to you — imploringly. A beggar on the street might look at you imploringly with his hand out. A kid might look at her mom imploringly when the ice cream truck goes by. This is a strong word for when people really need or want something and are trying hard to get it.