I agree that it is an exotic and romantic place but I would also contend that Shakepeare used it because the place name begins with Ill. There is an illness in Illyria.
Orsino is lovesick. He thinks he is in love with Olivia who does not return his love. Olivia's illness is her self indulgence in rejecting life while mourning her dead brother. Feste points this out. Malvolio's illness is self love while Toby's illness in his feelings of uselessness. The world has passed him by and he spends his time in idle pleasures.
Viola is the cure so to speak. Olivia falls in love with the disguised Viola who falls in love with Orsino who finds this new "young man" intriguing and quite knowledgeable about women.
By the end of the play, there is real love between Viola and Orsino. The relationship between Sebastian and Olivia is another matter since it is based totally on appearance.
Is everybody cured in Illyria? Not really. Malvolio storms off swearing revenge on everybody, but at least the Duke has been cured or perhaps more correctly he discovers what love really is.
Illryia is an actual location, found in the Adriatic coast of Italy. (For Shakespeare, the placement of his romantic comedy in this exotic locale had much to do with the audiences of his day. Those theatre-goers "liked to project themselves into the world of romance" as well as indulge in a "nostaglia for by-gone days." Not much has changed about human nature, has it? We still long for unfamiliar beauty, still pine for a the "good old days."
If you think about it, the warm Italian breezes must have been quite appealing to those in the comparatively dim, rainy, and grey weather of England. And chivalry? That most by-gone of by-gone eras, even by the 1500s? Well, many then, as now, cherish the thought of its renewal, at least for a few hours of escapism.
Shakespeare, Wiliam. ed. Bruce R. Smith Twelfth Night. Bedford-St. Martin UP. Boston: 20001.
It is very exotic, beautiful and romantic place, that's why Shakespere placed this play in the Greek country. Actually, this location is a real place, located at the western part of today's Balkan Peninsula, inhabited by the Illyrians. It gives a romantic feeling towards the place so that's why Shakespere choose this location for the play: romance is in the air