、、、。
The whole time, what the old man had said kept buzzing in my head, like the wing-beat of flies circling a melon.
Literal
That [genitive-の] between throughout, old-man [genitive-の] said-thing [subject-が], melon [genitive-の] around [object-を] fly-around flies [genitive-の] wing-sound [like-のように], head [genitive-の] inside [at-で] buzzingly was-ringing.
A vivid simile-heavy sentence, probably from literary fiction. Structurally it's a topic-less narrative: その間ずっと ('throughout that time') sets the temporal frame, then じいさんの言ったこと ('what the old man said') becomes the grammatical subject of 鳴っていました ('was ringing/buzzing'). Between them is a long simile introduced by ~のように ('like X'), which is itself modifying 鳴っていました. ぶんぶん is a 擬音語 (sound mimetic) for a buzzing, droning noise — bees, flies, engines. 蠅 (はえ, 'fly') is a kanji a learner might find rare. Note the 'path を' on 周りを ('circling around') — flies are flying a route around the melon. An atmospheric, literary composition.