You really do ask the key questions. If I could explain how consciousness interacts with the brain to produce a physiological response I'd probably win a Nobel prize. To my knowledge, the best minds are still working on this problem, and it's not strictly a matter of any one field. Nevertheless, the beauty of logic is that as you have deduced, on some level intention must be physical.
Researchers may never get to the bottom of the "how problem". By that, I mean that we can never get to the bottom of anything existential. There is always a point at which we cannot explain how something comes into being — it just does. The best we can do is plot the relationships between things and put them into their corresponding context.
For example we don't know how a conductive wire with a current running through it wrapped around a ferrite core produces a magnetic field. We just know that it does, and we've created all sorts of rules about the way the relationships between those things results in certain outcomes. But exactly why or how any of that happens on a fundamental level still boils down to the forces of nature, the existence of which remains unexplained — or at best theoretical.
So if I were to theorize on the answer to your question, I'd say that first of all, intention is a product of the brain. There's lots of evidence for this, the most convincing being that researchers have measured the neural activity of the brain when subjects are performing certain tasks, some of which are much like you describe ( simply acting to pick something up ), and the truth is that our brains start to formulate the action before we ever become aware that we are thinking it.
This is quite logical when you think about it. Fully formed thoughts ( like the intent to do something ) don't just appear suddenly in a flash fully formed. They start in a particular neuron, followed by other neurons, that follow pathways that link them all together, and at some point the result is enough connections to trigger our awareness and action. But quite literally, that decision is already made before we ever become aware of it.
The causality of intention is therefore somewhat illusory. But it's not quite that simple. The whole system involves a sort of feedback loop that takes place inside the brain in a structure known as the thalamocortical loop. I tend to favor the idea that the composition of our thoughts – what we typically think of as our conscious experience, is composed of EM fields created by the brain.
To get this across more clearly, I like to use an analogy. Suppose we have a house made entirely of bricks. Are the bricks the house? Or is the house the bricks? If the house is the bricks, why isn't a randomly arranged pile of bricks also a house? Now if we think of this house as where our thoughts reside, we can never look at a single brick and see the house, just like we can never look at single neurons and see consciousness.
But by organizing the parts in a certain way — and only in a certain way, we get the emergence of something that is greater than the sum of its parts. I hope this helps to some extent.
Oh, and while you're at it, please explain how physicalism equates with naturalism.
I made an attempt to relay the answer to that question a number of posts back. Suffice it to say, that there is no consensus among philosophers on this topic. I'm just in a particular tent in a particular camp. A good place to start looking for some related info is here - Stanford Encyclopedia of Philosophy
IOW, you can't explain. Instead, you employ diversion, obfuscation and hand-waving to support your preferred belief in physical monism.