You set the bias of the base of a transistor which makes it conductive or not(from emitter to collector). Eventually with a very small handful of transistors to 50 or so, some voltage dividers, and some other stuff you can get basic gate logic(and, or, not, xor, and similar). Then by combining those gates you can start to fake math, if at the same time you mangle around with out input and output works.

Then use some nifty tricks to do things like storing a charge in the capacitance or out put of the out put of transistors and you can start to have the ability to store a individual "string" of data(on and offs, 1 or 0). Mangle around some more gates and get an address system. then you start to use logic to mangle around with that output and input and you eventually develop a stack.

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So far from having a "true understanding" myself though. Was just a requirement to learn so that it wasn't "magic", but rather "beyond scope" when troubleshooting.