The expression often relates to someone's physical inability to wake up or respond due to exhaustion. After the marathon, he was out for the count on the couch.
Pro Tip 2/3
Not Literal
It's used figuratively, not implying an actual count out like in boxing, but rather a state similar to that. She was so tired that by 9 PM she was out for the count.
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Temporary State
This phrase suggests the person is temporarily unable to engage, not permanently. He's out for the count now, but he'll be up for dinner.