。「」。
She has reddish hair. That's why she's nicknamed 'Carrot.'
Literal
She [topic-は] reddish hair [object-を] is-having. [because of that-それで] 'carrot' [called-という] nickname [subject-が] is-attached.
赤味がかった is formed with the suffix ~がかった (tinged with), attached to the noun 赤味 (redness) — 'tinged with red, reddish.' This suffix works with colors and qualities: 青みがかった ('bluish'), 芝居がかった ('theatrical'). それで connects the two sentences as cause and effect: 'because of that.' The phrase あだながつく means 'a nickname sticks' — つく here is 'to attach/stick.'