Sometimes it's even translated, "Be under the authority of your husbands."
There is an old saying, and it's a true one: "All translators are liars." That's why there are three words used to describe one thing.
The word "submit" is "hupotaso," which means "to place under." It means to have an order to things. Otherwise we'd have chaos.
My take on this is that the phrase is defending "patriarchy." To me patriarchy means that men created everything in the world - society, technology, culture, and the ideas that support all of them. Some women of course have contributed to some of these things, but they are statistical outliers. It's not due to "oppression" but the fact men and women have different brains.
"Submitting" means "don't screw up things men have created/discovered, okay? Support them...don't try to destroy them or society."
Ultimately men are protectors/providers and women are nurturers (to be more precise, I think they are receptive/reactive). Not all, of course, but most.
Whether women believe it or not, all of them are under the "authority" of men. They are completely dependent on men, because either they can marry men or they can marry the State, which itself is "patriarchal." If men withdrew all that they have created from women, women would, as Camille Paglia noted, be reduced to living in grass huts.
When men, for whatever reason, allow women to stop being under their authority, then we end up with feminism, which is leftist, which itself what I call the Bad Feminine: destructive, seductive, not nurturing at all.
Left to their own devices women destroy everything when there are too many of them in one field. This is what happens when men allow women to do what they want, free from the authority of men.
Ultimately this is men's fault for allowing women to do this. You can see these warnings in mythology, when Adam listened to Eve when he was warned not to, and in the story of Pandora, who loosed everything bad on the world (the only thing left in the box was Hope).
Any woman with any sense (that is, ones without that inflated "self-esteem" based on nothing) knows what men have done for them, both good and bad. They know the bad things these days but most don't know the good. There is no gratitude or appreciation for any of the good.
While there is room for improvement in everything it's not going to come about through through leftism (the late Catholic anarchist Erik von Kuehnelt-Leddihn referred to leftism as "the overthrow of the Father"). In other words it's not going to come about through women, except the exceptional individual ones.
All you have to do is look at the 20th Century, the century of leftism, when monarchies (Kings and Queens, Fathers and Mothers) were overthrown. It's been estimated up anywhere from 177 million to 200 million died. And all because the Bad Mother overthrew the Good Father (and the Good Mother, too).
Not one monarchy went down fighting. Why? For mature constitutional monarchies, killing their subjects is like killing their children.
Now of course we can look around to today. It's easy to see what we have: a government of Bad Fathers and Bad Mothers (leftist, of course) and almost no Good Fathers and Good Mothers. No good can come from this and instead a whole of bad.
Which is what we are seeing right now.
Amazing what the past can tell us about the future, if we just listen.