・・・。
The snow is piling up in silent drifts... on my body.
Literal
Snow [subject-が] quietly-[adverbial-と] fall-pile-up... I [attributive-の] body [on-に].
しんしん is a 擬態語 for snow falling quietly and steadily — specifically the hushed, soundless piling-up of snow. (Notably, the mimetic conveys 'quiet' even though the word looks onomatopoeic — Japanese lets you hear silence.) 降り積もる is a compound verb from 降る ('fall') + 積もる ('pile up,' 'accumulate'). The stylistic centerpiece is 倒置法 (inversion): the locative に-phrase is pulled out to the end of the sentence after the main clause and a pause, creating a dramatic 'wait, it's piling up ON ME?' punchline. Writing オレ in katakana (instead of 俺) lends a casual, stylized flavor.