Doing something serenely means acting in a calm, peaceful way. Think of a vacationer relaxing in a beach chair, gazing serenely at the wide blue ocean and the expanse of soft sand.
When your cat is relaxed and content, he purrs serenely, and your levelheaded sister seems to continue smiling serenely no matter how stressed and anxious you are. This adverb goes back to the Latin word serenus, used to describe weather that's "peaceful, calm, and unclouded" and in a figurative way for people who are "cheerful, glad, or tranquil."