In literature, the eyes of females usually carry significance. The color of the eyes is meant to be a reflection of the soul. If the eyes are fair and kind, so is the soul of the character. That is the idea, anyway.
Black eyes are, therefore, often compared to a dark soul. A piece of common knowledge in England, where Dahl is from, illustrates this very point. It is well-known there that a very interesting historical character was known for her large, black eyes: Anne Boleyn, the second and super controversial wife of Henry VIII. This is same wife, the second of six, who eventually was decapitated and accused of everything under the sun, including witchcraft. It is said in England that King Henry should have watched out by simply looking at how dark her eyes were.
The dark nature of the eye color impedes the light of the sun, metaphorically speaking, "the good," "the truth," or anything positive that is represented by light, to go through them. Therefore, the idea behind representing "black eyes" is to point out that there is, or could be, something sinister lurking beneath the darkness. Something dangerous.
Therefore, to have letters "like black eyes" essentially means that the "Bed and Breakfast" sign that Billy stares into is hypnotizing him and luring him to come inside the house of the landlady.
BED AND BREAKFAST, BED AND BREAKFAST, BED AND BREAKFAST. Each word was like a large black eye staring at him through the glass, holding him, compelling him, forcing him to stay where he was and not to walk away from that house...
As we already know, Billy ends up entering the bed and breakfast, which is a home so comfortable, homely, cozy, and warm that is impossible not to be enticed by it. As we also know, the sign did carry its purpose: it lured him inside, only for him to meet his end shortly after.