hi! there's room for more input on this, but you may be interested in these...
What was life like before bras? - featuring /u/cecikierk and /u/colevintage
When did people start wearing underwear? Did men and women start to wear underwear at different points in history? And when did underwear start being something the majority of people wear? - featuring /u/colevintage
When did it become socially unacceptable for women in western societies to not wear a bra? Is there any documented reason why this came about? - more from /u/colevintage
For clarification, are you asking about when women began to wear bust-supportive clothing or about the origins of the modern brassiere?
Short answer: the change in undergarments followed trends of outer fashion. For instance, in the 1910s when the flapper style became popular, the hour glass figure that the corset gave a women was no longer in fashion. Instead, a more "boyish" figure of a straight line was in fashion causing the undergarment companies to follow the trends of flapper dresses. (both books listed below cover this idea in detail).
For one of the best books just on the patents of modern bras you should see Uplift: The Bra in America by Janet Farrell-Beck and Colleen Gau. The book is easy to read and traces the evolution of bras that modern American women wore based on the patents that bra companies filed.
And for a history on lingerie in general in American, including sexuality and colors of lingerie, see An Intimate Affair: Women, Lingerie, and Sexuality by Jill Fields.