The main way is by averaging error terms toward zero.

For example, Da Vinci's machine cuts the new screw blank in the centre of the carriage, which is driven by leadscrews on either side. The nuts would have likely been something like a wide strip of leather clamped over the screw thread, so there's enough compliance to average over a few pitches of thread, and the position of the cutter would be close to the average of the position set by each leadscrew thread.

Imagine the rough-cut screw threads have a pitch vs rotation angle described by p1(θ) and p2(θ). Running the machine then creates a new screw which is nearly a duplicate of the drive-screws in the machine but with pitch p3(θ) = (p1(θ) + p2(θ))/2. You can make two of these screws and swap them for the two leadscrews in the machine (it's built to be easy to do this). The random errors from a rough-cut screw gradually average out. But the cleverness doesn't end there. You then flip one of the screws backward end-to-end, so now you're averaging p3(θ) with p4(L-θ). You can also offset θ by any amount for either screw by offsetting the change-gear and re-clamping the carriage nut. Repeating these actions, you gradually can eliminate all systematic thread errors from the initial rough-cut screws and converge towards cutting a screw with nearly-constant pitch.

(It doesn't end there either; there's really a lot of flexibility with Da Vinci's design. Changing the gear ratio lets you create a fine-pitch thread from a coarse-pitch thread, or vise-versa, or cut a multi-start screw by rotating the blank 180 degrees or end-over-end).

The traditional way of making a multi-start screw is to use a dividing head and rotate the part an appropriate angle. If you're working with change gears, choosing a gear that's a multiple of the number of starts could provide alternate way to do this.