I've come out at a few workplaces, the first time I ever did it I just told my boss "I'm having a sex change, this is my new name." Lol. Not the most tactful description but he was old and French and wouldn't have understood me if I described it in a more complicated way. He just asked if it would affect my job, if I needed time off etc, I said "no, I just wanted you to be aware, I'll get you the paperwork when I have it." He said no problem, that he was a proud Canadian and that the meaning of living in Canada was freedom to do whatever you please and not be discriminated against. Then he said if anyone gave me problems to let him know and he would deal with them.
The second time I came out to work, I had changed my name already but wasn't on hormones so looked female, and someone referred to me as a she. I basically just said "I'm not a woman, I'm going through a transition so please refer to me as male." They were cool about it too and corrected people who misgendered me that didn't know.
The third time, name changed and I was stealth but the old name showed up on a background check. I basically just said to my boss "you're going to see on my paperwork that I used a female alias a few years back. Just to clear up any confusion, I no longer use that name." She literally just said "OK" and it changed absolutely nothing.
Short and sweet is the best. The smaller a deal that you make of it, the less other people will care. Just be as nonchalant as you can about it, all you can do is relay the information, after that the responsibility is on them to respond appropriately and treat you with the respect you deserve.