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The teacher sticks their nose into other people's business at every opportunity.
Literal
Teacher [topic-は] something [quotative-と] say-[when-と] person [attributive-の] matter [at-に] mouth-put-out-does.
Two set phrases to notice. 何かと言うと (literally 'when one says something') is an idiomatic 'at every opportunity / for every little thing / any chance they get.' 口出しする ('to put out one's mouth') is 'to butt in, to meddle, to interfere verbally' — note the verbal compound with a noun 口出し + する. 人のこと ('other people's business') is a fixed phrase for matters that don't concern the speaker. Subject of 口出しする is the same topic 先生 — it's a general habitual statement.