When she was young, she refused to eat anything fattening.

Literal

She [topic-は] young time, fatten things [topic-は] nothing eat [tried-ようとしなかった].

ふとるものは何も食べようとしなかった is a strong negative: she wouldn't even try to eat anything that would make her fat. ~ようとしなかった (didn't try to / refused to) uses the volitional + としない pattern to express refusal or unwillingness. ふとるもの (things that make you fat / fattening foods) uses a relative clause where ふとる (to gain weight) modifies もの (thing).