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My understanding of this statement (and it has simply developed organically through practice in the Church - I cannot quote a definitive answer from anyone) ... is that laymen, just as monastics, are responsible for entering spiritual battles with the passions, developing virtues, resisting temptations, praying regularly, doing good to others, and all the rest that goes into forming us into the likeness of Christ (by the grace of God).
I don't think it is intended to say that a layman should keep the exact same prayer rule or fasting rule, attend as many services, etc. as a monk any more than it is saying that a monk should have children and sacrifice himself for the spiritual development and good of his family. It is not in the particular details that we are exactly the same, but in the overall goal and, where applicable, to the general responsibilities, though perhaps not the exact method or quantity of practice.
There can be an idea that "only monks struggle with the flesh/demons" and the rest of us are somehow "off the hook" for these efforts, and I believe that is what we must realize is really the same for both of us. Whatever life a person finds themselves in, they are to seek God and be formed in His image through that life.
I hope that makes sense. :) Lord forgive me if I've made any error there.