The meaning of Sheriff Bell's dream was that he was ashamed of himself because his fear of death prevented him from doing his duty.
In other words: Ed knew the criminal was in the motel room, but his fear of death prevented him from doing his job, which, in this case, was to die honorably in the line of duty.
He interprets this moral failure as a desecration of the legacy of his father.
This is contrast with the moral failure of Anton: Anton killed the woman who refused to play his coin game, which forced him to assume the moral responsibility for murdering her. This incident is immediately followed with Anton's car accident, which represents him being punished by God and forced to rely on the charity of others for survival. This is why Anton demands the kid take the money in exchange for the shirt - because he recognizes that he's being punished (i.e. he recognizes the charitable symbol and that he doesn't deserve charity) which is an indirect admission of guilt.
He interprets this moral failure as a desecration of the legacy of his father.
This is contrast with the moral failure of Anton: Anton killed the woman who refused to play his coin game, which forced him to assume the moral responsibility for murdering her. This incident is immediately followed with Anton's car accident, which represents him being punished by God and forced to rely on the charity of others for survival. This is why Anton demands the kid take the money in exchange for the shirt - because he recognizes that he's being punished (i.e. he recognizes the charitable symbol and that he doesn't deserve charity) which is an indirect admission of guilt.