"Burning bridges" is an idiom for "destroying a relationship permanently." For example, if you quit your job and never want to go back there ever again, you might write a nasty blog post that explains why you hate everyone at that company. Once you post it and your former co-workers read it, you won't be able to work there again. You have "burned bridges" with that company and with your former co-workers. The idiom makes sense because if you burn a bridge, there's no going back.

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It is generally wise to not burn bridges with people, school, work, etc., because you never know if you might have to go back to them in the future.