"Mankind always finds a way."
The Common language evolved from the tribal languages of humans and halflings. Humans are a naturally gregarious and mercantile race, and so Common has grown throughout the Ages, filling with loan-words from Sylvan, Darro, and even Urgyn and Drakyr. Most cultures have had frequent exposure to humans in some way, so the Common language is the de facto trade language of the realms.
"Seheyá benlá lehes se heyá Eyladrim fa shì."
The Sylvan language evolved in the Feywild, as the natural tongue of the fairy races. The ancestors of the eladrin and elves taught it to the spirits and treefolk, and so Sylvan (along with Urgyn) is the language most often spoken by druids and shamans trained in the Old Ways.
"Urhkhas kh'rgh arhark !Khaar hyashk hrrukh."
The Urygyn language also evolved in the Feywild, as the natural tongue of the goblin races. While the two languages sound very different, they share similar structural syntaxes - indicating that they may have once had a common origin.
"Shehim hah Daru khazhan, an dûl ben jahmith shah."
The Darro language has no known origin. While many of its earlier 'loan words' are clearly taken from proto-Common (or possibly vice-versa), the language itself - like the Dwarves who speak it - seems to have arisen spontaneously with little outside influence.
"Ssshah hassh-ihah Drakh-ha hrrrauughh-ffassh! Hhaghh-k!"
The Drakyr language is a derivative of the elemental language of the Primal Chaos, and thus a descendant of the proto-language that forms the basis of magic and Creation itself. Both dragons and elementals speak Drakyr, although many learn other tongues from their dealings with the Material world.