How does Tom convince Aunt Polly that dreams can hold the truth?

Tom uses dreams to convey his feelings for Aunt Polly. He doesn't express or expose his feelings verbally, but rather, assures her that he cares for her by using his "dreams" to share his emotions. Tom knows what happened, he was there, and he knows he was wrong..... he admits to the "truth" of things by recounting a dream he never had.... a moment he witnessed.

“Well, I don’t say it wasn’t a fine joke, Tom, to keep everybody suffering ’most a week so you boys had a good time, but it is a pity you could be so hard-hearted as to let me suffer so. If you could come over on a log to go to your funeral, you could have come over and give me a hint some way that you warn’t dead, but only run off.”

“Yes, you could have done that, Tom,” said Mary; “and I believe you would if you had thought of it.”

“Would you, Tom?” said Aunt Polly, her face lighting wistfully. “Say, now, would you, if you’d thought of it?”

“I—well, I don’t know. ’Twould ’a’ spoiled everything.”

“Tom, I hoped you loved me that much,” said Aunt Polly, with a grieved tone that discomforted the boy. “It would have been something if you’d cared enough to think of it, even if you didn’t do it.”

“Now, auntie, that ain’t any harm,” pleaded Mary; “it’s only Tom’s giddy way—he is always in such a rush that he never thinks of anything.”

“More’s the pity. Sid would have thought. And Sid would have come and done it, too. Tom, you’ll look back, some day, when it’s too late, and wish you’d cared a little more for me when it would have cost you so little.”

“Now, auntie, you know I do care for you,” said Tom.

“I’d know it better if you acted more like it.”

“I wish now I’d thought,” said Tom, with a repentant tone; “but I dreamt about you, anyway. That’s something, ain’t it?”

“It ain’t much—a cat does that much—but it’s better than nothing. What did you dream?”

“Why, Wednesday night I dreamt that you was sitting over there by the bed, and Sid was sitting by the woodbox, and Mary next to him.”

“Well, so we did. So we always do. I’m glad your dreams could take even that much trouble about us.”

“And I dreamt that Joe Harper’s mother was here.”

“Why, she was here! Did you dream any more?”

“Oh, lots. But it’s so dim, now.”

“Well, try to recollect—can’t you?”

“Somehow it seems to me that the wind—the wind blowed the—the—”

“Try harder, Tom! The wind did blow something. Come!”

Tom pressed his fingers on his forehead an anxious minute, and then said:

“I’ve got it now! I’ve got it now! It blowed the candle!”

“Mercy on us! Go on, Tom—go on!”