How does writing to a C variable end up controlling hardware?

Through memory-mapped I/O: peripheral registers occupy fixed addresses in the CPU's address space, so a store instruction to the right address physically acts on the peripheral. In C you express that by casting the address to a volatile pointer of the correct width and dereferencing it, *(volatile uint32_t*)0x40020014 = 1, or, idiomatically, through a register-overlay struct (GPIOA->ODR = 1) whose members are laid out to match the datasheet offsets. The store isn't "saving a value" in the normal sense; the hardware decodes that address and, for an output-data register, drives the corresponding pins. The volatile qualifier tells the compiler the access has side effects and the value can change outside the program, so it must perform exactly the loads and stores written, in order, without caching or eliding them. That combination, a known address, a load/store, and volatile, is the foundation of every driver.