LGBT questions..

Sammie Jae • 🇩🇴 | Virgo | mommy of five💚
Okay, so in my home country being a transgender is something that takes A LOT of nuts. People will literally kill you for being so, & won't hire you. 
I've never had an experience with a transgender person until today, & I honestly think I insulted him/her because I'm so uneducated on the matter. & I've honestly researched it, like 10 minutes ago. & I'm even more confused now. 
Please explain to me how a transgender person isn't the sex they were born. 
& if it's a mental thing, wouldn't it be a mental illness? 
Note that I have nothing against anyone, I just don't understand! My country is nothing like the states. & I was taught one way about transgenders, but I'm honestly trying to learn about them correctly. & I'm not meaning to insult anyone, so please don't take offense. I'm trying to correct my ignorance. 
Thanks 😊
(Hopefully this makes sense.)