The reason you don't understand it is the definition of 'gender' people use for this argument is so nebulous that it has basically no meaning at this point. Add to that that 'social construct' is also rather nebulous and you got a recipe for a discussion that is completely useless as people have no idea what the other is really talking about.
As best as I am able to decipher this thinking it seems to come from a few ways of thinking:
1. That 'gender' is more akin to what is more commonly known as 'gender role'. Now gender roles are indeed pretty society dependent, although they are not completely so (which goes into the whole issue of what exactly is a 'social construct'). The problem here is gender role has never been a rigid structure with pretty strict boundaries like gender is generally considered to be, so interchanging the two concepts causes nothing but confusion.
2. That 'gender' is more akin to lifestyle, a vastly different concept than traditionally thought. Here is seems the wish to elevate their personal lifestyle to the level of gender is, well frankly grounded in a rather narcissistic need to seem more important and special. This, ironically, causes people to pigeonhole themselves more to seem more unique.
Then we have the more reasonable camp of:
3. That 'gender' is one's self image of their sex. This is the one most transgender people who actually go through procedures seem to go with and has the most scientific evidence. So essentially it allows gender and sex to be not aligned, but gender is still intimately linked to sex and in the vast majority of cases is aligned with sex.
And finally we have the die-hards in opposition where:
4. 'gender' is just an alternate word for 'sex'. There is nothing more too it.
As to what constitutes a 'social construct', in simple terms we think of this as 'is this a physical fact, or is this just something society arbitrarily decided on.' Problem is, that is a black and white fallacy. Honestly there is nothing that society hasn't got at least some play in, but there is also nothing that physical facts have at least some bearing in. So what constitutes a 'social construct' is itself a social construct.
The go to example for social construct is boys = blue, girls = pink, and yeah that's pretty much as social constructy as it gets, but even here there is biologically driven pressures to distinguish the sexes. It certainly didn't need to be blue and pink, didn't even need to be color, but it is very likely we'd always have had some distinguishing feature between children that we would have assigned. On the opposite end, you may say 1 + 1 = 2 is about as absolute as it gets, but even here we could have done the same equation as 1 1 + = 2, or + 1 1 = 2, or I + I = II, or numerous other representations. None of them change the core principle, but still it ends up that 1 + 1 = 2 can be seen as a social construct... which is also why the term is mostly useless as it doesn't really tell us anything of worth.
