what makes you uncomfortable, specifically, about your trans clients? is it that they seem to wear the "wrong" clothes?
lots of good answers on here already. i think it's great that you are trying to understand better.
this may seem a bit crazy if it's the first time you heard it, but gender isn't our sex, nor our chromosomes. society has made it that way - if you have a penis, you're a man. if you have a vagina, you're a woman. but in history, there are plenty of examples of gender not being about body parts. native american tribes often had a third gender (neither woman/man), or were two-spirit (both man and woman, or a mix of the two). it's european and western philosophy that took over and made gender out to be based on body parts/chromosomes.
that being said, try to imagine yourself "trapped" in the wrong body. try imagine hating everything about you that makes you manly. a lot of us feel that way. a lot of us have tried really fucking hard to be okay with how we were born, but nothing worked...until we transitioned. transition, and often hormones, will make a lot of the pain and suffering go away.
however, this whole "being trapped in the wrong body" isn't for everyone. for me, i simply want to be seen as a man. but i don't mind having my body parts (i did remove my breasts, though). my body is mine, i wasn't born in the wrong one, but i think society should stop expected me to be a woman. because i'm not. my not having a dick doesn't make me any less of a man - do men who lose their dicks in war or accidents suddenly expect to behave like women? do men who were born without a dick, or had to have surgery to fix a "birth defect" dick, suddenly become women?
also, keep in mind, there are many many people who were born with both - dicks and vaginas. many people have both XX and XY chromosomes, in some combination. so even "biological" sex isn't clear cut. but because society treats us all like shit, not a lot of people know about intersexed people. and most trans people in modern western society have only ever been beaten, harassed, or treated like freaks until recent history. but that's not how we've been treated in many other cultures before europeans came to the americas. nor in many other cultures worldwide.