The other answers have explained (excellently!) the origin of the ellipsis symbol, and how older chat programs displayed typing indicators. However, this doesn't directly answer the question.
To my knowledge, the first mobile app use of the "three dots in a bubble" indicator is iMessage. As Samuel pointed out in his comment, this was taken from iChat on the desktop, which had this UI as early as 2005.
When iMessage launched, the "typing" indicator (and read receipts) was a big differentiator from SMS. A lot of other mobile apps have since copied the typing bubble UI, even though desktop chat apps have had these types of indicators for a while.