IPA: /ˈblɪðəˌsoʊmli/
KK: /ˈblɪðəsəmlɪ/
In a cheerful and carefree way; showing happiness and joy.
She danced blithesomely at the party, spreading joy to everyone around her.
Blithesomely is formed from "blithe" (from Old English "blīðe", meaning happy or cheerful) and the suffix "-some" (meaning characterized by or full of) combined with the adverbial suffix "-ly" (meaning in a manner). The word describes an action done in a cheerful or happy manner.
Think of 'blithe' meaning happy, and remember that 'blithesomely' means doing something in a happy or cheerful way.