The short version (which is not very short, sorry!) is this:
First you need to understand that sex and gender are not the same thing, despite both being referred to as male and female. Sex is biology and gender, well, gender is a couple of things based on who you talk to.
When you talk to trans activitists or the non-binary community you will be told that gender is self expression, gender is your "self" effectively. It's whether you feel/know you are male or female (or something in between). The majority of people have a gender which matches their sex at birth. This group refer to those people as being "cisgender". But some people's gender does not match their sex, these people would be transgender. Transgender people may or may not want to use hormones and surgery to change their sex, but usually they wish to legally be recognised as male/female based on their gender rather than their birth sex. Then there is another group of people who feel that gender being restricted to "male" and "female" is restrictive and unnecessary. They argue that gender is not a binary, it is a spectrum. People can be a mixture of male and female, they can change depending on the day, they can have a completely different gender, or they can have no gender at all. But overall, gender (whether binary or nonbinary) is considered the important one over sex because it is self defined and hence more true to a person than their sex which they cannot choose (although this is where I get a bit confused because surely you also can't choose your gender, either.) This group celebrate gender as they feel it is an important part of a person's identity and feel that smashing the gender binary will lead to more acceptance of gender nonconforming behaviours.
Talking to radical feminists will get you a different perspective. Gender is not self expression. Gender is a set of restrictions and expectations placed upon a person based upon their biological sex. Hence, gender is external, not internal, considered harmful and to be abolished, not celebrated. Radical feminists believe there should be no limits placed upon a person because of their sex except for biological limits, e.g. only biological females can carry and bear children, only biological males can produce sperm.
The clash comes partly because one group wishes to abolish gender while the other wishes to increase its importance over biological sex. But, strangely, both groups are supportive of the idea that your sex or gender should not influence what opportunities you have, what interests are available to you, or basically anything about your personality or life except when it comes to medical issues. It's just they disagree on the way to achieve such a utopian state. Trans activists believe that expanding the definition of genders and celebrating non-standard gender configurations will open people's minds and blow away old stereotypes and eventually we'll get to a point where people are accepting of different genders and individuality. Radical feminists believe that complicating and celebrating the idea of gender is counterproductive, that talking about gender as though it is a concrete thing is reinforcing harmful stereotypes and ideas and that instead we should be talking about people, not gender.
Then another problem which radical feminists have is that because transgender activists wish people to be able to use the labels of male/female, boy/girl, man/woman (etc) in relation to people's gender rather than their sex, this is confusing as in radical feminists' beliefs, gender does not matter, only sex does (and sex matters far less often than we use it). Feminists find themselves barred from discussing issues such as women's health involving ovaries and wombs, because the word "women" is expected to include transwomen (who do not have ovaries and wombs), and excludes transmen (who often do) and non-binary people who are biologically female. It makes it very difficult to talk about issues which affect biological women when suddenly the word woman does not mean the same as it used to do. That might not be a problem, if equality was already here, but it is not.
The secondary issue with this is that anybody who gender identifies as female is allowed entry into woman-only space. This throws up questions about the reasons we need women-only space to begin with. The issue feminists have is that woman-only space is created to protect biological women from the violence and dominance of biological men. It is nothing to do with identity and everything to do with safety. By the way, I don't think many people have a serious problem with toilets. I think most people understand that trans people just want to use the toilet like everybody and are not going into toilets with the aim to rape or assault women, it's more that toilets represent a gateway for universal entry into places which should be women only, such as women's prisons, sexual assault and abuse centres, women's violence shelters, and so on.
Hopefully that is helpful. I did skim read the thread so have probably repeated some points but I didn't see a summary in the same format.