if you just want to kick the tires, there are two pre-built firmware images in the 'files' section.  Pick one of them (USB CDC or UART).  You can burn this with whatever flashing tool you have, e.g. I use the "STM32 ST-LINK Utility" with a ST-Link-V2.
There is also a way of doing this over the UART with the built-in bootloader, though I've never done that myself.  Google "Programming an STM32F103 board using its USB port (Blue Pill)" and you'll find some instructions if you want to give that a whirl.

The project is built with "System Workbench for STM32", which is a free toolchain from ST Microelectronics.  You'll need to install that.  The source code is complete and you should be able to build out-of-box except  note that I also hacked the Open OCD config file to allow me to use the 128 KiB flash.  (there's a post on doing that)
Then you can burn the image you built with whatever.
You might be able to alternatively use PlatformIO -- it looks like it can import the STM project, but I've never tried this myself.

The firmware presents a command-line interface over whichever serial mechanism you have chosen as per build options or pre-built firmware image.  The interface is described in detail in 'manual.txt' in the files section.
Note that the firmware by default will operate in 'physical' mode -- i.e. it expects an actual SP0256-AL2 to be connected.  You can change the mode to 'simulated' which will allow you to play with the synthesizer with just the BluePill alone (well, you still need your LPF and audio amp).  If you do this, you probably will want to 'persist' the settings so that the choice will survive a reboot.