Like everyone else, I was shocked by this. Why the fuck would you ban transgender people from joining the military? That's like banning women, mexicans, bisexual people - the fuck? However, a little research explains why. Basically, the military has to fund expensive gender reassignment surgeries for soldiers currently undergoing transition. I don't believe it is the place of the military to fund any kind of expensive surgery; the US healthcare system should take care of that. (Luckily for us Brits, we have the NHS).
Additionally, trans people have higher levels of anxiety and suicide rates than non-trans people. I don't think a person who is physically or emotionally crippled in any way should join the military. I know I certainly wouldn't because I'm in recovery for alcoholism and I have depression and suffer from panic attacks. Not being in a position to give the best possible service means it makes sense to not join.
BUT - I think if a person has already transitioned and is on hormones, and are physically and emotionally healthy, then there is nothing wrong with them joining the military. That goes for anyone. I'm not pro-war and think the government uses too much of taxpayers money funding the fucking military and nuclear weapons, but that's another topic. If a trans person is already serving in the military and is not in an emotionally unstable state, then discharging them because they are trans is appalling.
http://dailysignal.com/2017/07/26/why-forcing-the-military-to-pay-for-sex-changes-would-be-disastrous/
https://heatst.com/life/gender-reassignment-cost/
http://edition.cnn.com/2017/07/26/politics/trump-military-transgender/index.html