This phrase is derived from the idiom “children should be seen and not heard.” Which means that children are not important and should not interrupt “grown up” talk by participating. Here she is making that same parallel to children by using the saying. It does suggest the idea that she has no contribution to make to “real” conversation. So you are correct that the not heard part refers to her opinions. However the meaning is more that her opinions are neither important nor valid.
It is key to note that she does not believe this and is merely saying what her society thinks of her.
This phrase can also be used in a positive manner. It can be used to say that you would rather observe than force your opinions on others. When used in this way it is usually done in the first person.
Ex: Interns in my company are usually seen and not heard.
Ex: I think sometimes it is better to be seen and not heard.