A short answer to that would be: nobody knows for sure!
The Buddha was born as Prince Siddhartha, in Lumbini, and grew up in Kapilavastu, modern day Nepal. Going by the recorded cultural practices of that time, might not the Prince have worn the chunky ear ornaments favored then by men? In time, these heavy jewelry pieces would have resulted in long earlobes. This is perhaps the most prosaic explanation available.
There is also a symbolic significance to the Buddha’s elongated earlobes. In Eastern cultures, large ears are associated with wisdom and revered by others (think Lord Ganesha and in more modern times, Mahatma Gandhi). These are ears that are big enough to listen to all our tales of suffering. Magnanimity and compassion therefore are also qualities linked to such physical features and the Buddha was certainly the embodiment of these virtues.
Finally, there is perhaps, a message to Buddhists in those ears. Every human is a potential Buddha; as such we should remain open to the suffering of others.