The 'belief' is that sex is important. As a society, we could either:
a) define the boundaries of the men/women categories purely by biological sex. This is the GC position - that biological sex is important
b) define the boundaries of the men/women categories loosely on biological sex, but with anyone allowed to switch categories if they want to. This is the TRA position because they believe that biology is less important than how the individual identifies
These are two different possible ways to categorize humans. GC people and TRAs have different beliefs on which is preferable.
GC position
The GC position is that the purpose of laws and social practices which treat men and women differently is to mitigate the disadvantages that women experience as a result of our reproductive role and biology (eg vulnerable to sexual predators because evolution has favoured different reproduction strategies for men and women, due to the different biological cost of reproduction; less physically strong than men because in women evolution has favoured adaptations to support and survive pregnancy and childbirth, and in men evolution has favoured adaptations that help them impregnate women). 'Discriminatory' laws and practices exist to enable women to take a full and equal part in society.
GC people believe that if you let people switch categories, then the laws and social practices don't work to allow women to participate equally in society. So this proposed new way of categorizing men/women is really regressive for women.
TRA position
The TRA position is that they want to remove social restrictions on how people behave which they consider unnecessary and mentally harmful. They see this as progressive because they are only considering the effect on Trans people, not the effect on women.
A final thought
The TRA position is that they want to keep the man/woman categories, and they want all the same laws and social practices to stay in place - but using the new categories with 'everyone allowed to switch'
The GC position is that where sex doesn't matter - e.g clothes, hair, jobs, interests - then we shouldn't be treating men and women differently anyway!
If there is a legitimate reason to treat men and women differently, then it has to be based on reproductive class in order to work.