How was Mrs. White able to make the inference that her son was dead?

Initially, Mrs. White doesn't quite comprehend what she's been told about Herbert, but we can infer she understood he was dead only moments later.

The visitor bowed in assent. "Badly hurt," he said, quietly, "but he is not in any pain."

"Oh, thank God!" said the old woman, clasping her hands. "Thank God for that! Thank—"

She broke off suddenly as the sinister meaning of the assurance dawned upon her and she saw the awful confirmation of her fears in the other's averted face. She caught her breath, and turning to her slower-witted husband, laid her trembling old hand upon his. There was a long silence.