、。
She refuses to leave her room, taking pains not to catch another cold.
Literal
[topic-は] again cold [object-を] don't-catch [so-that-ようにと], deliberately room [from-から] go-out-[try-to-ようとしない].
Layered modal sentence. 風邪を引かないように ('so that she doesn't catch a cold') uses the negative form ~ないように with the purposive ~ように — 'so as to not.' The と after ように has a hint of the quotative — captures that the speaker has taken this stance in mind. あえて ('daringly, deliberately, intentionally') is a stiffer adverb that flags a chosen, conscious decision — often one slightly against expectation. 出ようとしない is the negative form of ~ようとする ('try to do') — 'does not try to leave,' i.e., refuses to leave. The whole sentence captures someone holed up at home guarding their health, perhaps overcautiously.