This is a great question, but it needs to be reversed: why do fairytales seem set in an idyllic medieval period?

What we think of as fairytales are published and abridged versions of folktales (German Märchen). These were the oral novels of pre-modern Europeans who tended to place these stories in a romantically-conceived former time This tended to be a medieval setting when the stories were collected in the nineteenth century. Because folktales typically ended happily - "Happily ever after" (as opposed to legends, stories told to be believed and usually ending horribly) - folktales captured the medieval setting in a positive way. So the answer to your question, is that the oral literature of pre-modern Europe depicted an early period - which happened to be generically medieval - as the setting for stories of heroism and of good always winning against monstrous opponents.

This sort of romantic depiction of a past time is not uncommon. It's how the "Wild West" became much the same sort of place of heroism and great deeds. Similarly, thirteenth-century Icelandic saga writers depicted the wild "Viking" times of several centuries before in an ideal, heroic way. Reality is, by nature, much harsher. People of every period lived their lives ignorant of the technology that we assume is essential to one's well being, and medieval people lived, laughed, cried, and died just like we do. Whether you or anyone would really like to be thrust back to the middle ages - or the Wild West or the time of the vikings - can be summed up with one word: antibiotics. You might be doing very well in that former time until you developed a tooth abscess. That would tend to take the shine off of anyone's sojourn into the past.

"The Middle Ages" was not solely a European phenomenon, although the fairytale-ish-ness was not solely whimsical singing in the forest and feasts in the castle either.

I agree with /u/itsallfolklore re tooth abscesses, but if we're doing a Rawlsian Veil of Ignorance about whether you'd want to live in the Middle Ages, also bear in mind that you might end up as an Iraqi peasant whose fairytales are more like the Arabian Nights than the Knights of the Roundtable.

In which case of course, there are no forests, but you probably have access to the largest and most glamorous city then-extant in the world, Baghdad.

Although personally I've always preferred the Maqamat of Hamadhani to the Arabian Nights: http://www.sacred-texts.com/isl/mhm/index.htm