The story
By figurative extension from the concept of being literally out of breath (from talking so much, exerting oneself so much, and so on), which may sometimes cause one's face to look wan.
Etymology from Wiktionary, CC BY-SA 4.0.
until one is blue in the face
Today it means: For a hopelessly long time; used especially in respect to discussing, explaining, insisting, etc. Full definition →