Remember the sabbath day, to keep it holy. (Exodus 20:8)
We should fear and love God that we may not despise preaching and His Word, but hold it sacred, and gladly hear and learn it. (Martin Luther)
I think in this explanation Luther focused narrowly on a particular problem of his day (also a problem now, IMO) and thereby captured only part of what this commandment means. I see two purposes in this commandment.
One is that humans, over time, do more, better, if they get regular days and times of rest. God programmed times of rest into ancient Israel's religion and culture, including the weekly Sabbath. “Rest” sounds so formal and sterile! What do you, reader, find restful or, dare I say it, fun? Taking a nap? Reading a book? Playing tennis? Taking a hike? Writing an article like this? “Rest” is as individual as we are! God really did and does care about His people, including their/our physical and mental well-being!
The second is that God knows that we need regular times devoted to our relationship with God. If we are busy with our subsistence 24x7x52 – work, eating, family, sleeping - that consumes all of our time, physically and mentally. Our relationship with God will shrivel to almost nothing.
Jesus and Paul changed the focus for Christian believers from observing a specific day to every day belonging to God. Most Christians observe and have their church worship services on Sunday, but the New Testament does not command this. Christians need to reflect on the state of their relationship with God and ways of growing it. Regular worship services are good, IF one is truly engaged in what is being done – the praise and worship, the prayers, the preaching and teaching (Luther's concern). But that is just a start! The kind of fellowship Christians need in their lives goes far beyond sitting next to a fellow Christian once a week or 20 minutes of chatting over coffee and cookies or a potluck supper, milling around and eating with scores or hundreds of people. Real fellowship - in which lives and concerns are shared and people's spiritual gifts can be discovered, developed and exercised – happens in small, stable, groups that meet regularly. And our relationship with God needs alone time – with God's word, in prayer, listening to Him. Take time for rest; take time for your relationship with God and your fellow Christians.