Kobolds originally learned the Draconic language directly from true dragons. The earliest kobolds were instructed to speak Draconic in a variety of accents and dialects. When kobolds became independent of their dragon masters, they took the Draconic tongue with them, dispersing it among their various clans and tribes. As these independent kobolds interacted with one another, their inherited Draconic dialects mingled, creating Yipyak, the common kobold tongue as it's known today. Kobold sorcerers know how to speak true Draconic, which requires more exacting pronunciation than the dialect that kobolds commonly speak from day to day.
The runic script of the Draconic language was originally developed by kobolds, as dragons have little need to write. It originated with cave paintings, then became pictograms, and finally the Draconic script used today.