Map processor to circuit diagram
The problem is that in order to understand how the computer gets to 1 + 2 = 3 you have to understand about 2 levels deeper than you've gone. 

Roughly a computer is organized (in terms of fields of study) like this from highest level of abstraction to the most physical reality:


Application Software
Virtual Machine
Operating System
BIOS
Embedded Systems
IP Blocks (Sub-units/Peripherals)
Logical Blocks
Gate-Level
Transistor Level
Semicoductors
Device Physics


To properly understand why the computer can produce 2+1 = 3, you must first decide what you are willing to accept "on faith" and what you will not believe until you internalize it. That piece of information will be at the level two below the thing you understand. So if you want to understand an adder circuit at the logical level you will need to understand the basics of "digital" transistors (specifically CMOS).

Using your earlier site as an example, consider this resource. It discusses the "Full Adder" -- the minimum completely general purpose circuit capable of addition/subtraction including carry-in and carry-out.

You will also need to understand how numbers are represented in 2's complement (the number system used in modern computers for integer arithmetic).

If you really want a world-class introductory course, I cannot recommend Professor Scott Wills at Georgia Tech highly enough. He passed away last year of cancer, but his course lives on. The Georgia Tech ECE2030 (introduction to computer engineering) class has its text book and exercises all online.

Good luck!