Paul is talking about authority and how accountability is on the account of the one with authority/dominion. He has a spiritual implication when he says that.
We first need to know what SIN is and it all starts in the garden of Eden. Sin is not an action but rather it is the spirit of lie in someone's mind/spirit. From the mind (indeed the expression of the mind) it is manifested through spoken word or action. Even though most actions are usually called sin, they are actually manifestation of sin (spirit) and spirit is passed from the one with dominion to their subordinates.
So Eve was the one who sinned first, yet sin (spirit) and there death came to the world on the account of Adam because he had dominion over everything/world. If Adam had not fallen for the lie, sin and therefore death would not have entered the world.
This is shown in the fact the Eve ate the forbidden fruit but her eyes were not opened until the her partner in marriage (because they are one flesh) who had dominion/authority also ate. As soon as Adam eats, both their eyes are opened. It shows that Eves sin was put on the account of Adam.
Therefore, sin (spirit) whose payment is death was put on Adam but each had to still pay for their actions/manifestation of sin. Adam was to eat in his sweat and Eve was to labor during child birth.
In the same manner sin is defeated through authority/dominion when the church (bride) marries God (Jesus)- her sins are put on the account of the one with authority (Jesus) who pays for it as long as we remain faithful/obedient/submissive. And because we are in marriage and we are one flesh with the one who overcomes death/sin, His righteousness is put on our account as the faithful submissive bride (Church).
Paul was very symbolic with that passage- the church is in labor because we still have to pay for our actions and we have to undergo persecution by the antichrist. At the end of it, the church will bear a son (the kingdom of God). This will only be true if we remain submissive to Christ.
This below, is an understandable version of the same:
Eph 5:
21Submit to one another out of reverence for Christ.
22Wives, submit yourselves to your own husbands
as you do to the Lord.
23For the husband is the head of the wife as Christ is the head of the church, his body, of which he is the Savior.
24Now as the church submits to Christ, so also wives should submit to their husbands in everything.
25Husbands, love your wives, just as Christ loved the church and gave himself up for her
26to make her holy, cleansing
b her by the washing with water through the word,
27and to present her to himself as a radiant church, without stain or wrinkle or any other blemish, but holy and blameless.
28In this same way, husbands ought to love their wives as their own bodies. He who loves his wife loves himself.
29After all, no one ever hated their own body, but they feed and care for their body, just as Christ does the church—
30for we are members of his body.
31“For this reason a man will leave his father and mother and be united to his wife, and the two will become one flesh.”
c 32This is a profound mystery—but I am talking about Christ and the churc