I've been trying to find books&other scientific material that mentions why the gender equality is important and should be achieved. Any help would be greatly appreciated.
Philosophy is about reasons. If you want research, then you are looking at anthropology, economics, sociology, psychology, etc.
Gender equality in the workplace has hugely increased the number of workers, that alone has seen unequal societies that don't depend on oil or mineral wealth largely become backwaters. Eg 'Gender equality boosts economic growth'
In the geological record, high sexual dimorphism is associated with greater extinction risks New driver of extinction: Adaptations for sexual selection & Sexually dimorphic swallows have higher extinction risk. This makes sense, because investing in plumage and performances, limits resources for survival. If both genders of a species are similar, presumably they are both closer to the optimum for their niche.
This great Mindscape episode: Episode 22: Joe Walston on Conservation, Urbanization, and the Way We Live on Earth makes the case that the issues around peak human population and it's impacts on the environment, can be alleviated more than anything, by more female education, family planning, and antenatal care.
So, evidence-based economic, geological, and environmental arguments. Now, philosophical arguments. The defining quality of advanced intelligences is intersubjectivity, the capacity to imagine the perspective of others, and invite them into our own, which also correlates with tool use - 'seeing into' objects. Discussed here Is the Categorical Imperative Simply Bad Math? :)
I was just looking at another example of this: Bigotry and the human–animal divide: (Dis)belief in human evolution and bigoted attitudes across different cultures
It's like how many of the arguments against slavery did not focus on the slaves, but on the impact of maintaining slavery on those doing it. Similarly with the banning of most animal fighting bloodsports, the concern was as much or more what kind of people it made, as on saving animals from cruelty. Intersubjectivity helps us understand this, by drawing divides between races, genders, species, we limit the scope of our world, and the basis for cooperation with other beings. By sharing our experiences with others, our shared reality is expanded.
It's fair to ask given all this, what benefits arose from gender inequality. Fundamental drivers are differential investment in offspring. Modern drivers have been the shift from self-sufficient homesteads to having workers, which in pre-industrial times took a worker at home to support a worker away from home. In times of increasing industrialisation there were often increased pressures on gender-role conformity.
Women After All by Melvin Konner summarizes a lot of scientific research about the differences between the sexes and, specifically, reasons why including more women in positions of power is good for the world. (It pays respect to the spectrum of gender identity while focusing mainly on the biological characteristics known to be due to chromosomes.)
Not specific to gender, but social contract theory (Hobbes, Locke, Rousseau) requires every citizen, or at least a very large part of them, to (figuratively) sign the contract.
Assuming we adopt this rationale for rights, the best way to achieve universal adoption is by guaranteeing equal rights for everyone. If privileges are granted without a solid reason, dissent will arise. What is more, when everyone has equal rights there is an incentive for everybody to help their fellow citizen resist oppression, because by defending their neighbor's liberties they defend their own.
In the case of gender, privilege on one side automatically alienate half of the population, which naturally tend to restrict difference in rights to the strict minimum required by physiological disparities (support for pregnancy, etc)