A friend of mine at my school's pride club asked me the "why be trans" question last year. (Ha, he thinks I'm a member because I'm gay, which I may be, but not in the direction he thinks!) Anyway, it was, I assume, a lot easier to explain to someone who's genuinely interested and already familiar with LGB issues. I explained to him the two sides of things as I understand them. I said something along the lines of "Some folks just don't feel at home in their bodies, a condition known as dysphoria, and feel that having bodies of the opposite biological sex would make them more comfortable. Additionally, some people feel that they fit better into the gender roles assigned to the opposite gender of what they were assigned at birth, and they want to be seen and treated as a member of that gender by society."
Shayne, there are some points I'd try to make to your ex if you really want to try to educate him, but don't forget the old saying "you can bring a horse to water, but you can't make it drink"; some people are so stuck in their way of thinking about something that they'll resist all evidence against them and bury their heads in the sand. If he doesn't understand that being a man is about more than the downstairs plumbing and genes, you'll have to start from scratch on explaining what gender is in the first place.