Why did Sor Juana Inés de la Cruz write her reply to the Bishop of Puebla?

Sor Juana Inés de la Cruz's letter to the Bishop of Puebla was a response to his own pseudonymous letter criticizing her for her attention to and concern of earthly matters and quest for knowledge. This was considered very unbecoming for a woman at the time. Sor Juana's letter is a defense of a woman's right to education and the pursuit of learning. She uses her letter to showcase her intelligence in an age and place where women were considered less intelligent and capable of knowledge than men. She even provides a framework of how a woman can properly learn so as to improve society as a whole.
Cleverly, she opens her letter by stating that she is not as smart as others may think, but then she shows her wit and wisdom by incorporating language and ideas throughout the piece that highlight her intelligence. Her letter responds to the Bishop's arguments and shines a light on her own abilities at the same time. Throughout the letter, she deconstructs many of the patriarchal practices of her society and promotes a woman's right to an education. Sor Juana Inés de la Cruz's letter is thus seen as an early work of feminism.

Sor Juana Inés de la Cruz (1648?–1695) was a well-educated Mexican woman who was renowned for her knowledge and literary ability. Seeking to further her knowledge, she entered a convent and remained there until her death. Her study of non-religious subjects made her a target of criticism in both political and religious circles.
In 1690, a person masquerading as "Sor Filotea de la Cruz," who was actually the Bishop of Puebla, Manuel Fernandez de Santa Cruz, published, without Sor Juana's approval, a letter that she had written criticizing a sermon. Sor Filotea also took Sor Juana to task for the secular nature of her studies and reading. In her famous response, Sor Juana defended her right as a woman to gain an education and stated that her secular studies made her more able to understand the scriptures. Her scholarly and eloquent letter is regarded as among the first defenses of a woman's right to an education.