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She spoke slowly so her students wouldn't miss what she said.
Literal
She [topic-は] students [subject-が] her [possessive-の] words [object-を] miss [would be bad if-といけないから] slowly spoke.
聞き逃す is a compound verb: 聞く (hear) + 逃す (miss/let escape) = 'to miss hearing.' ~といけないから means 'because it would be bad if' — a grammar pattern expressing precaution. The whole reason clause (生徒たちが言葉を聞き逃すといけないから) explains why she spoke slowly. The politeness of ゆっくり話しました contrasts with the relatively complex grammar.