To cut against the grain means to run a sharp knife at an angle to the muscle fibers running through a piece of meat. The idea here is to shorten those fibers as much as possible so chewing them is a breeze.
According to Mr. Kapusta, you can cook a steak perfectly, juicy as can be, but if you don't carve it against or across the grain to shorten those muscle fibers, then it won't be as tender.
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