How does memory get stored physically? It varies depending on the computer. Several different methods have been used, from ultrasonic pulses in columns of mercury to charges stored in capacitance wells in semiconductor gates.
Currently, I think that bits are stored inside the CPU in the state of latches in registers, made of a couple of CMOS transistors, and in RAM (dynamic RAM) in said capacitance wells. Google DRAM.
Other technologies used include magnetic cores and static RAM.
Current microprocessors are hugely complex, but the basic premise is that most everything works with a clock, like a metronome in music. Transistors are turned on and off to allow bits to move from one element to another (adders, multiplication units, shift registers etc.) It's a bit like watching people move through turnstiles at a railway station, but imagine that the turnstiles are all synchronized so that they all open and close at the same time.
There is surely a whole lot of material online if you look. You used to be able to find datasheets for Intel processors, but they may have given that up. You can probably find schematics for simpler processors like the 8080 or some RISC processors, which are available as macros to include in your designs in ASICs or big FPGAs.