In the phrase "the scales have fallen from my eyes" why did they use the word "scales"?

According to the Oxford English Dictionary, this expression comes from the Bible:

Taken (after Acts ix. 18) as a type of that which causes blindness (physical or moral).

Etymologically, it comes from "scale" in the fish sense, with the first example of "scales" being used like this coming from the 1382 Wycliffite Bible or Cursor Mundi (a1325-a1400) according to the OED. For more information, see the freely-available Middle English Dictionary.

As for what exactly is being described here, Ellicott's Commentary for English Readers on Acts 9:18 says:

The description suggests the thought that the blindness was caused by an incrustation, caused by acute inflammation, covering the pupil of the eye, or closing up the eye-lids, analogous to the "whiteness," that peeled (or scaled) off from the eyes of Tobit (Tobit 11:13). Like phenomena are mentioned by Hippocrates, and the care with which St. Luke records the fact in this instance, may be noted, with Acts 3:7; Acts 28:8, as one of the examples of the technical precision of his calling as a physician.

Elsewhere on BibleHub it says that the original word used in Acts 9:18 is "λεπίδες", which is defined as:

a scale, a scaly substance thrown off from the body.

Further to Laurel and Michael Harvey's answers, the Online Etymology Dictionary refers to scale as

"skin plates on fish or snakes," c. 1300, from Old French escale "cup, scale, shell pod, husk" (12c., Modern French écale) "scale, husk," from Frankish *skala or some other Germanic source, from Proto-Germanic *skælo "split, divide" (source also of Dutch schaal "a scale, husk," Old High German scala "shell," Gothic skalja "tile," Old English scealu "shell, husk"), from PIE root *skel- (1) "to cut."

In reference to humans, as a condition of certain skin diseases, it is attested from c. 1400. As what falls from one's eye when blindness ends (usually figurative), it echoes Acts ix:18 (Latin tanquam squamæ, Greek hosei lepides).

One can presume that Saul had scale-like incrustations on his eyes.