Got a few years? Sheesh, what you're asking is a large subject (or there's a smaller question in there, somewhere that you want to ask but don't know how ;)
In my defense, it's true, the binary state of a "set of physical components" _is_ conveyed to other sets of physical components via wires (or conductive paths).
If you want to know how those binary states (i.e. 'on' or 'off') are stored/manipulated, then you need to understand _flip-flops_. And, to understand flip-flops you need to understand Transistors (at least transistors in their non-active states. Most microcontrollers (as per your example) use MOS transistors, so you might start with those.
If what you're asking is how information is transported and interpreted and made to do cool and interesting things (as well as dull and boring things -- depends on the application), then you need to understand binary encoding and protocols. Another fairly large subject. You might also need to know a bit about logic, multiplexing, and perhaps even binary arithmetic.
If you are asking about machine code (i.e. how a program manipulates the "silicon" within a microcontroller (,etc)), then that yet another large subject.
BTW, there's also a third state (besides 'on' and 'off') called the _float_ or _high-impedance_ state (though that is more a Von Neumann architecture thing). But, that propels you into the realm of open drain transistors and pull-ups.
So, rather than one of us trying to give you a 10-unit crash course in computer electronics, I suggest you take a night class or get a book and then hit us with specific questions as they arise.