Reading and writing are forbidden for a number of reasons:
First, women are forbidden to read and write because the government of Gilead does not want them to be able to communicate with one another secretly. They cannot send one another letters or messages that they might use to start or join a resistance. Since they are deprived of all modes of communication besides verbal speech, the women's communication can be much more tightly controlled, because it can be monitored: overheard, caught on video, and so on.
Second, reading is one major way that people educate themselves. When we read, we learn about other people and other societies, and we expand the way we think about the world. The Gileadean government does not want women to be educated or thoughtful. They are supposed to be obedient and submissive. In fact, if you think about it, it is only the first generation of women in Gilead who will be literate. After them, if no women learn to read or write, they will be so much more docile and compliant because they simply will not know another way. They will not be able to read stories about heroes who stand up to injustice, and therefore, they might not even have a way to identify injustice when they see it.
Third, by keeping the Bible—which Gilead claims to take as its foundational text—secret, the government can control consumption of it. The government can prevent people from reading inspirational stories that might compel them to action and only allow citizens to come in contact with the stories that would seem to support their society. Further, they can make changes to the Bible, some subtle and others not so much, in order to suit their own end—Offred is aware of some of these changes and points them out to us.