يسلموا اديك is used when someone gives or hands you something from what I understand, like a plate of food or drink and means "may your hands be safe", but is a way of saying thank you. Maybe some others can add a more complete answer as far as the use of يسلموا on its own.
It's شنباتك shanabētak, where shanabēt is a synonym of شوارب shawēreb "mustache". (Some other dialects use 'singular' شنب.) Then ممنون mamnūn is the passive participle of/an adjective derived from the verb من manna, so alone its original meaning is like "grateful; thankful; indebted", from which it became an interjection meaning "thanks".
The full expression, ممنون شنباتك mamnūnshanabētak -- which, put simply, would mean "thank your mustache" albeit not as an imperative -- definitely strikes me as cutesy rather than as a serious expression of gratitude. You'd use it jokingly with a friend or someone close, but perhaps not as a general "thank you".