But what is 'feeling like a woman'?
Fortunately for myself, I am a cis woman. My body doesn't feel wrong, female pronouns don't make me feel uncomfortable, I've never experienced dysphoria. For that I am extremely lucky. It's not really a picnic even once you take out the discrimination.
Being cis does not, and has never, meant that I exactly fit the gender stereotypes attached to a "woman". Gender stereotypes are BS. If I got a buzz cut and threw out all my skirts and got a high-ranking office job, I wouldn't stop being a woman. Being a trans woman is the same; it means that they identify as a woman, not that they identify as someone who wears high heeled shoes and likes pink. They have the strong conviction that they are indeed a woman, even if their sex doesn't correlate with that.
As a cis woman, I can't tell you how it feels to have that conviction, because I don't know. A trans person might have a better answer as to what it feels like, but there aren't many on Mumsnet, oddly.
All I do know is that I will never be able to feel how they feel, so it's stupid for me to try and understand. All I can really do is exercise my human decency; be compassionate and respectful and when they want to talk about it, listen.
That probably doesn't help you at all when it comes to a direct response to your question. But I can tell you that when I was still struggling with the concept of being trans or genderfluid or agender, my turning point was realising that your question doesn't really matter that much.
I would never be able to understand what they felt like, and that's just the way it is. The ability to realise the world is wider than your own experiences is an important one, and someone else's gender will always be outside of your experience. The fact that I do not understand their experience of gender means that my understanding is limited, not that their experience is invalid. I cannot say that their gender is not real, because I cannot feel or understand it.
Maybe you'll get some good answers on here about what it actually feels like. If not, I daresay trans people have posted about their experiences elsewhere online.