A few ways. This particular project is doing it by hand and very tedious.
The traditional way of transplanting large trees while keeping the root system intact is with a hydrovac. A machine the size of a jet engine that liquifies the soil with water and then vacuums it up. [1]
More recent developments have tried using an AirSpade which doesn’t use water but compressed air to blow apart and then suck the soil without making a slurry which is better as the soil can be redeposited in the same hole rather than discarded[2]