I have owned since 1975 a copy of an earlier, hardcover, Pantheon reprint edition of this superb collection, which was originally published in 1945. I have used it for both light reading and for serious study (while in courses on Baltic and Slavic Folklore and Folktale Studies). The selection and translation of stories both seem first-rate. (For the latter, I have had to rely on the opinions of those who actually read Russian, instead of just having studied it in school.) The accompanying illustrations are properly enchanting -- and only occasionally are placed where they give away the point of the story.
The only real drawback is that it is still merely a selection from about three volumes (depending on the edition you prefer) of "skazki." This is the Russian term for oral tales of marvels, adventures, and misadventures, equivalent to the German "Maerchen." In both cases, the English term "Fairy Tale" is the conventional, but not really adequate, translation. (As usual in large collections, only a handful of tales concern anything like fairies.) One of the requirements for the selection seems to have been that the tales chosen should be acceptable to American parents in the 1940s, but otherwise the considerable variety of the original seems to have been largely preserved. The suggested reader age of "9 to 12" conceals the pleasure that adult readers with interests in folklore or Russian culture will derive from the volume. Fortunately, they may be lead to it by the fine supplementary material at the end, although this is now half a century old.
Afanas'ev (various transliterations) was one of the many nineteenth-century collectors inspired by the Grimms,. By most accounts he was one of the most responsible, even though his practices of recording and documenting texts are hardly up to modern standards. (Neither were those of the Grimms, for that matter.) The main collection from which this was excerpted was the sourcebook for Vladimir Propp's "Morphology of the Folktale," a key work in modern folktale studies, but as Roman Jakobson (yes, the Structural Linguist) points out in his commentary to this collection, the book had already established itself as a gem of Russian literature, an inspiration and resource for poets.