You can download and inspect the code that implements this demo. Although the demo is for an RTOS the code used to place the CPU into a sleep mode is the same whether an RTOS is being used or the application is running on bare metal.
There are generic things you can do to place a Cortex-M3 core into a low power state (see the WFI instruction). To get extreme low power then you have to do chip specific things as well. The above linked code performs some chip specific pre-sleep processing (turn of peripherals, set the chips own sleep mode, etc.) before calling WFI, then does some chip specific things when it returns from the WFI instruction.