You could use a Gecko drive (any of them with a step/dir interface) but they would be overkill for this application. The NEMA-17 size motors he's using wouldn't require the power that the Gecko drives are capable of.
You would be better off with a RAMPS board (~$50) and the associated A4988 drivers (~$10/ea). That is what he's using in the video.
I already have NEMA-23 motors and my machine is bigger than this one.
Can you tell me if there is a ADURINO board that can run such combination ( NEMA-23, Gecko drive G540 )?