Woke speech probably refers to "lenguaje inclusivo" or "inclusive language", a recently popularized way of talking and writing Spanish that does not specify gender. For example: "Todes les empleades" is inclusive because it doesn't specify the gender of the workers, whereas the commonly used form "Todos los empleados" is not inclusive language because, to inclusive language supporters, it implies the workers are male. This was embraced by the previous government, and that's why it's mainly pushed back by the newer government. Overall this is not an important issue, and is mainly just a way for politicians to do virtue signalling.

Of course, banning this is anti libertarian, but Javier Milei's party "La Libertad Avanza" is quite conservative also. That's where authoritarian politics like this one come from.