Submission means that there is a mission, and in order to fulfill that mission, we must bring ourselves under authority or submit (just like an army). The ultimate authority is God. God put man and woman, husband and wife, on earth to produce righteous seeds.
This begins with the man's submission to God. The man must humbly obey his creator through the grace and power of the Holy Spirit. If he circumvents this submission, he is setting a bad example for his wife and children.
A woman's submission is no different. She submits to God, and then, to her husband, who is her earthly head. A wise woman will honor her submission to God by being submissive to her husband as long as what he asks does not defy the word of God.
When a man or woman refuses to submit, they disqualify themselves from being a viable partner in a relationship. Submit means I am ready and willing to serve. Marriage above everything is a platform to serve (a very powerful truth) so, a man or woman who rejects submission, are unsuitable as a marriage partner.
"Let this mind be in you, which was also in Christ Jesus: Who, being in the form of God, thought it not robbery to be equal with God: But made himself of no reputation, and took upon him the form of a servant, and was made in the likeness of men: And being found in fashion as a man, he humbled himself, and became obedient unto death, even the death of the cross." (Phil 2:5-8)