This is the 14717th most frequent Russian word.
песнями
The Russian word "песнями" means "by songs" or "with songs" in English (instrumental plural form of "песня").
In this sentence, 'песнями' functions as an instrumental plural noun, indicating the means by which the action is performed (with songs).
Они развлекали нас народными песнями.
They entertained us with folk songs.
Here, 'песнями' is also in the instrumental case, showing the source or medium of inspiration (by the songs).
Я вдохновляюсь её красивыми песнями.
I am inspired by her beautiful songs.
In this case, 'песнями' is again instrumental, denoting the method or accompaniment of the greeting (with songs).