I think it might be because I really hated my own body for so long. It took a long time to realize I was encountering sexism, but fatphobia I knew about before I had a name for it. I can remember being pre-puberty
and lamenting that I was "thin in one direction but not the other" (those were the words I used at freaking <10 years old) - I realize now that I'm thin everywhere but at the time I was terrified of being even a little bit fat.
Fatphobia hurts thin people too (but I try not to say that too much because I get annoyed when people feel the need to too emphatically bring up how sexism hurts men). And I felt the effects of fatphobia first. I have a RIGHT to love my body, damn it, and so do people with more adipose tissue than me!
I think the FA movement also uses more direct words to express ideas that apply to all -isms. The unqualified acceptance of the self isn't danced around - or so it seems to me, having first met the concept via Kate Harding. In fact, I am using my familiarity with the ideas of FA, now, to come to a form of mental/emotional self-love via therapy. Being able to love oneself no matter what is a central, CENTRAL concept in FA, and it's one that strikes close to home and makes me want to write essays that are beyond my actual writing skill.
I hope this makes sense. >.<