I had this analogy come to me this morning as I meditated. Think of this stilling of the mind like that of a mother holding the hand of a troubled child. As the child sits and stews, fussing and worrying his mind over this or that that troubles him, his mother reaches out and touches his hand. Suddenly the child is calmed, because his focus has turned from his mind, his troubled thoughts, to the love and presence and assurance of his mother that all is well. It is in simple trust of that love, that the child lets go and his thought 'empties' into that of his mother's love and assurance. As he does this, he breathing slows, his body relaxes, his mind stills.
This is meditation practice. It's no more "hocus-pocus" than what we all do with our minds in common things. The only difference is that it is a deliberate practicing this calming of the mind in a disciplined practice, rather than going through our lives flaying around hoping to find moments of peace through this or that distraction. As we practice this placing of the mind into a state of silent "trust", as it were, we soon learn that the rest of what goes on in the mind is none other than what we do to ourselves through giving into the thoughts as the truth of reality itself. It's all an illusion of the mind, things we tell ourselves to be true, believe in them, and then respond to them emotionally, psychologically, and spiritually. All that meditation does is teach you to "be still, and know that I am the Lord". It is to learn to rest in Spirit itself as our Center, and our Ground, rather than simply learning to cope with the illusion of the mind and its storylines it creates for us.
How, and why, anyone objects to this, is beyond comprehension. Do they fear letting go of all of that garbage we swim in as we trying to keep our nose above water? Meditation teaches you, shows you, makes it apparent, you can actually stand up in that stream and step out of it, walk on water, as it were. "Peace, be still", Jesus said to the raging seas.