In Chapter 19 of To Kill a Mockingbird, what about Tom's voice was swaying Scout's opinion of him, and how did he fix the old door?

The fact that Tom's voice was quiet, "with no hint of whining," made Scout want to believe what he was saying.

Atticus had taught his children to be perceptive. Scout recalled that he "sometimes said that one way to tell whether a witness was lying or telling the truth was to listen rather than watch," and as Atticus questioned Tom Robinson, Scout applied her father's teachings. She noticed that when Atticus asked Tom if he had "ever set foot on the Ewell property without an express invitation from one of them," he responded three times in the negative. Although Scout thought that by repeating himself three times, Tom was protesting too much, she still found herself inclined to believe him because of the quality of his voice. Tom spoke "quietly, with no hint of whining in his voice," indicating to Scout that he was telling the truth.

Tom did not actually fix the old door, because there was nothing wrong with it. Mayella had asked him to come inside, telling him, "I got somethin' for you to do in the house...th' old door's off its hinges an' fall's comin' on pretty fast." Tom asked Mayella for a screwdriver, but when he looked at the door, he "pulled it back'n forth," only to find that "those hinges was all right." Mayella had lied about the door needing fixing to lure Tom Robinson inside the house, where she hoped to get him to make love with her (Chapter 19).