As I understand it, the Boring Company's strategy is to dig smaller tunnels. Since the volume of earth that must be moved scales with the square of the tunnel radius[1], a tunnel that is half as wide only requires a fourth of the dirt to be moved.
I believe this accounts for most of their estimated speed/cost advantage - dig thinner tunnels and move less dirt. They also seem to be coupling that with more continuous tunneling and less labor.
As far as I know, they're doing this with existing tunnel boring machines, just used differently.
Volume = pi * radius^2 * length