The act of ironing is arduous work. When you iron you must stand to do it; you must be careful of how you do it so as not to miss a part or to iron in a worse wrinkle than already exists. When you iron you are trying to "perfect" the item of clothing -- sometimes the item is easy like a plain, flat shirt; other times, the clothing is complicated with pleats, or other little details that make the task much more demanding. In the end though, there is a certain satisfaction with knowing that the ironer has mastered the clothing item and it is indeed better for the effort put into the task.
What I think is interesting about the story is the last line -- the speaker says she doesn't want to iron out her daughter like this dress upon the board. She doesn't want to iron out -- or make perfect -- her daughter. She just wants her daughter to be who she ends up being. The connection of the symbol of the iron to parenting is very interesting.
Ironing was no easy task at the time that this story is set. In fact, ironing has been one of the most enslaving, demanding, and dangerous jobs throughout the history of home economics. All you need to do is do a search on "the history of ironing" online and you will see how the tool itself has been modified hundreds of time for that same reason.
Moreover, the main character is "standing there ironing" setting the record straight and explaining a situation the same way that she is setting the clothes she is ironing straight. It is basically a symbol of being at the bottom of the totem pole: You are there ironing, slaving over something, sacrificing yourself.
That is the basic premise of it all.