。
She said she'd be turning 20 next year.
Literal
She [topic-は] next [possessive-の] year [at-で] 20-years-old becomes [quotative-と] said.
20 (二十歳, hatachi) marks an important threshold in Japan — until 2022 it was the age of legal adulthood, and even after the change to 18 it remains culturally significant as the age for the second 成人式 ('coming-of-age ceremony') and the legal drinking and smoking age. The で here marks the time at which the becoming will happen — 'in / at next year.' The reading 二十歳 ('hatachi') is one of Japanese's irregular age readings; most ages use ~歳 with regular sai readings (一歳, 二歳, 十歳), but 20 takes its own unique reading.