The Hebrew word here for "be still" is raphah. It means things like:
"Be slack" (re. your arms).
"Be idle" (stop working and striving).
"Take leave" (of all your busyness).
"Cease struggling" (with God).
Even, "be weak" (so God can be strong).
"Chill out, and know that I am God."
"Let go, and know that I am God."
Ps. 46: 10 does not instruct us to:
"Overwork, and know that I am God."
"Be busy, and know that I am God."
"Multi-task, and know that I am God."
"Try harder, and know that I am God."
"Get uptight, and know that I am God."
"Shop, and know that I am God."
We can draw a line connecting Psalm 46:10 with John 15, where Jesus tells us to "abide" and "rest" and "remain" in him, like a branch is connected to a vine. The promises for us include: Jesus will give us his peace, his joy, and reveal to us all the Father has made know to him.
This is the incredible invitation to "be" with God, with the seminal Jesus-idea that being is prior to doing; viz., being-with God precedes all relevant "doing" in the name of God.
The "be still" thing is a heart-thing. You don't have to be in a physically quiet place to have a still, quieted heart. A "stilled" heart can be had in the midst of outer chaos. Jesus himself walked into many dark, chaotic situations. We don't see him freaking out and taking on the outer chaos in his own, inner heart. There is a focused calm about Jesus that's all about his ongoing abiding in the Father. How does he do this? Who wouldn't want to live like this? How is this possible? In John 14-15 he gives us the answers.
Jesus says: "Anyone who has seen me has seen the Father. How can you say, 'Show us the Father'? Don't you believe that I am in the Father, and that the Father is in me? The words I say to you are not just my own. Rather, it is the Father, living in me, who is doing his work."
Then, Jesus gives his disciples, and us, this amazing invitation: "Anyone who has faith in me will do what I have been doing. He will do even greater things than these, because I am going to the Father." The Father and Jesus the Son will come and "make their home in us." (John 14:23)
If God is in your "home" (= heart), struggling ceases, and stillness remains. If the Lord is my fortress, and in his fortress I dwell, what shall I fear? Who wouldn't want to live like that?
Be still before the Lord today by:
1. Taking 10-15 minutes and just "be" in God's presence. For no purpose other than to "be still, and know that God is God."
2. Doing this again tomorrow. And the next day..., and so on...
3. Not striving to make something happen. You might say to God, "I want to know how to be heart-still, that I might know You better."
5. Watching, over time, how God meets you in that secret place, which is your heart.
There might be so much chaos in your life right now that you are thinking, "I can't find time to do this with all the stuff that's going on!" Or: "I could never get still before God with all the clutter in my heart!"
I can tell you from personal experience, from much time doing this, and from teaching this to many pastors and leaders over the years, the following: "Being still" will seem counter-intuitive in our crazy world of incessant, irrelevant "doing." But it is precisely what is needed, and what God calls you to do. In the phrase "know that I am God," it is important to understand that "knowing," in Hebrew, is a relational, experiential thing.
Psalm 46:10 is a "call to relationship" with God. As are the last earthly words of Jesus in John chapters 14-17.