。
The recommendation letter from the section chief made all the difference.
Literal
Section-chief [from-から] received recommendation-letter [subject-が] things [object-を] said.
物を言う (ものをいう, 'to speak / to say things') is an idiom meaning 'to be effective / to carry weight / to make the difference.' 課長 (かちょう, 'section chief') is a mid-level management title. 推薦状 (すいせんじょう, 'letter of recommendation') is a formal compound. からもらった ('received from') modifies 推薦状 as a relative clause. The idiom implies the letter was the decisive factor — it 'spoke' on the person's behalf.