In relation to talking to your daughter, the way I explained it to mine was:
It is totally unacceptable to exclude anyone from anything based on their gender. Gender relates to masculinity and femininity. Society has a tendency to label jobs, clothes, behaviours etc as masculine and feminine and then expect males to conform to the masculine ones and females the feminine ones. This is unnecessarily limiting for everyone and shouldn't happen. All people should be able to be as masculine and/or feminine as they wish. Transgender people can experience all sorts of issues because of their non conformity to society's expectations of male = masculine and female = feminine, and this isn't acceptable. We should promote tolerance and acceptance of gender non conformity.
Gender shouldn't be conflated with sex. Sex differences exist according to biology and, in some specific instances, people should be treated differently according to sex (e.g. medical screening, separate sporting competition, and sex segregated service and spaces for situations where women are vulnerable).
As this segregation is according to sex is not exclusionary and it does not discriminate against trans people. No one is excluded from their own sex category because of their gender identity. However, there is an overlap with issues with gender, because where people are gender non conforming, they can feel vulnerable because they don't present themselves like most people of the same sex. Here, again, we need to challenge and extend societies expectations around gender conformity.
There are also some people who deeply wish they were of the other sex or believe they are of the other sex. This is gender dysphoria and is a medical condition for which they should receive treatment and support