the software developers didn't use those methods for creation of the devices. The devices are a series of electronic switches. The developers had to talk to them in a language the microprocessors understand -- binary. But the algorithms they used for those devices are the same kids are taught.
Did you even understand why these algorithms worked at the time you were taught them?
Why they "worked?" They worked because they couldn't do otherwise! And you don't need to understand WHY they work anymore than you need to understand Bernoulli's Principle to take an airplane flight. If, however, you don't know that they exist and the batteries run down on your calculator, you are helpless.
In detail, this is a schematic of a 4-bit adder which can add two numbers, 0-15 and produce a result from 0-31. The symbols perform the logical operations AND OR XOR on two inputs producing an output and a carry.
This is a drawing of one of the four bit chunks. The different colors represent either conductive metal traces or different doping of the silicon to get the charge to move in the desired fashion.
The whole thing (all four bits) ends up looking like this. I think it's pretty cool. It helps to understand the math beneath it though to really appreciate its beauty.