Tom Robinson has a withered left hand that has been rendered useless. Since the Ewells claim that Tom has tried to beat Mayella, grabbing her by the neck.Mayella testifies,
"He got me round the neck, cussin' me an' sayin' dirt--I fought'n'hollered, but he had me round the neck. He hit me agin an' agin--"
Atticus Finch points out Tom's handicap in order to disprove the possibility of Tom's having been able to inflict the reported injury upon Mayella, and to discredit the testimony of the witnesses. For, it would be physically impossible for Tom to have choked Mayella and the marks on Mayella's face reveal that the bruises were inflicted by a left-handed person because they are on the right side.
With such proof as this, a defendant should be acquitted of the charges. However, Tom is not; thus, his conviction is clearly made by the bias of the jury.
The previous post by MWestwood was right on the money. However, another reason that Harper Lee gave Tom Robinson a crippled arm was to gain sympathy for him--at least from the reader. However, because Tom was black, he received little (or no) sympathy from the jurors and/or many of Maycomb's citizens. Readers probably felt a great deal of sympathy for Tom and many would hope for his acquittal for this reason alone. Jem and Scout, as well as many other open-minded townspeople, may also have had their heartstrings tugged a bit because of his condition.