What actually happens under the wall
Your foundation wall sits on a footing, and the footing sits on soil. If you simply dug the basement floor lower, you would be digging away the soil that footing is standing on. Underpinning replaces that soil with concrete before it is removed — but only ever in pieces small enough that the footing can bridge the gap on its own.
The perimeter is divided into numbered sections on the structural drawings. All the sections marked 1 are dug, formed and poured. Once that concrete has cured, the sections marked 2 open. Then 3. At no point is a section opened next to another open section, which is why the wall above never loses its support.
The last step in each section is the one people never see and the one that matters most. Concrete shrinks slightly as it sets, so pouring to the underside of the old footing leaves a hairline gap that carries nothing. That gap is filled with a stiff, almost dry mortar rammed hard into place — dry-packing — so the new pin is in genuine contact with the old footing and actually takes the load.