Elon Musk and top-level scientists from his neuroscience startup Neuralink Corp., who are developing a next-generation brain-computer interface, unveiled what they billed as a significant advance toward a therapeutic device Tuesday night.

The device would connect human brains and machines with more precision than other available devices, according to the company, which has been developing the technology for roughly two years. Neuralink is putting together a submission to the U.S. Food and Drug Administration to start testing...