What does it really say ?
A woman shall not wear that which pertaineth unto a man, neither shall a man put on a woman’s garment; for whosoever doeth these things is an abomination unto the LORD thy God.
It would be simple to brush away Deuteronomy 22:5 altogether, saying that it is part of the Law and therefore no longer relevant to modern Christians. After all, cisgender Christians, whose sex and gender are in alignment, no longer adhere to this prohibition either; women wear pants, men wear kilts, and clothing styles are more influenced by culture than religion. Rather than simply ignore or dismiss the verse, it is more useful to try to understand what the verse means in the context of the culture in which it was written.
First, even among Jewish rabbis, there is no agreement as to what this verse means precisely; there is, however, general agreement that it does not refer to those who cross-dress or to transgender people who wear clothing more aligned with their gender identity. The three predominant possible meanings of this verse have one commonality: wearing the clothing of the other sex with the intention to deceive. One interpretation proposes that the phrase “that which pertaineth to a man” refers to a sword or other pieces of his weaponry. In other words, a woman should not be given weapons and sent to war. It is also similarly suggested that this verse means that a man should not dress in the clothes of a woman to hide among women, particularly during a time when he should be soldiering.
The final two predominant arguments are in line with the general instruction found throughout the Jewish Law to keep things separate: types of seeds in a vineyard (Deuteronomy 22:9), an ox and a donkey when plowing (Deuteronomy 22:10), and wool and linen in clothing (Deuteronomy 22:11). Men and women in ancient Israel (and in some sects even today) were regularly segregated, thereby limiting their interaction. This argument is supported by the discourse on marriage and adultery that follows Deuteronomy 22:5 in Deuteronomy 22:13-30.
The intent of the law, in this last interpretation of the verse, is to prevent men and women from mixing by deceitful entry into the segregated space of the other sex with the intent of committing adultery. In the verse, adultery is what is called an abomination unto God. Cross-dressing is not strictly and always forbidden in the Law; it was specifically permitted beginning in the 16th-century Code of Jewish Law that allowed men to dress as women and women as men for the Jewish feast of Purim for the purpose of celebration, as opposed to with the intent to deceive and commit adultery.
The interpretations of the verse are diverse, and while many conservative Christians have tried to use the verse to condemn transgender people who wear clothing opposite to their birth sex, most Jewish legal discussion of the verse centers on the intention of deception.
Deception, however, is not the motivation of transgender people wearing the clothing of the gender with which they identify. When transgender people wear the clothing of their internal gender, they are trying to decrease their emotional and mental stress by bringing their outward appearance into congruence with their gender identity.
 Rabbi Jon-Jay Tilsen, “Cross Dressing and Deuteronomy 22:5,” Congregation Beth El-Keser Israel, http://www.beki.org (Accessed September 4, 2014).
 Rabbi Sylvia Rothschild, “With Increasing Joy, We Explore Our Dark Side: Purim Thoughts,” rabbisylviarothschild, March 2. 2014, http://rabbisylviarothschild.wordpress.com/2014/03/02/with-increasing-joy-we-explore-our-dark-side-purim-thoughts/, (Accessed September 4, 2014).
 “Cross Dressing and Deuteronomy 22:5.”
I highly recommend “Transparently” by Lisa Salazar.
As we all know, sometimes it takes a village.