I came out as transgender over a year and a half ago, I've been perceived as female by others for almost a year, and I've been on hormone replacement therapy for almost eight months. And, to risk sounding like a cliché, I'm starting to feel like a different person from who I was before; my perspective on life has changed so much that it's becoming increasingly difficult to relate to my older self. The way I think of myself, the way I relate to others, the way others relate to me... my thoughts about the future (something I'm finally not terrified of), and what it means to be a good person or a happy person or the sort of person I want to be. But feeling like a new person after a period of substantial change isn't a feeling unique to people who transition.
In fact, for quite a few months now I've felt like my "transition" has turned into more of a general ongoing self-improvement project. My gender dysphoria has lessened to a massive degree and allowed me to finally feel connected to the world in a way I wasn't before. It's like I've stopped being a robot and everything feels real for the first time ever. I feel actual motivation to work on myself, to mature and grow as a person, rather than just a desperate scramble for anything to make the pain stop.
Obviously HRT has been a significant factor in making me feel better, but there's lots of other factors too. Some of them are luck -- my genetics made "passing" (being perceived as female by others) a non-issue as soon as I got over my fear of expressing myself that way -- but lots of other things I can attribute to myself. Of course there are the so-called "superficial" things like the way my voice sounds (something that triggered a lot of dysphoria in the past, thus something I worked on a ton), but I think even more important has been my introspection. I have mainly female friends and spend time in female spaces (on the internet), and this has caused a lot of reflection about what it means to be female, the diverse experiences of womanhood, and what exactly this all means to me.
Basically, I've started feeling like a valid member of my gender instead of a wannabe. It's not super uncommon to hear trans people try to teach other trans people how to feel this way, but in my experience it rarely seems to have much effect. Getting over your internalized transphobia is something that, like all significant personal growth, has to come from within. And it's easy to trick yourself into thinking you're there when you really aren't yet.
I definitely have a lot of personal growth left to do, but nowadays I'm confident that I'm on the right track. Transition is definitely the greatest decision I ever made, and I've become a much, much better person. Life finally isn't hell! And growing breasts is pretty cool too :P