No, it's not a way to encourage population growth. Shame on you for thinking that. Archimedes' Screw is a way to draw up water for drinking or irrigating fields.
So the zombiepocalypse has passed. You now live in a world full of technology that there is no longer infrastructure or manpower enough to use. You pass the shut-down nuclear power plant. When you see the massive tech company buildings, you don't stop, because you know all that's in them is discarded ironic fedoras and iPads not sturdy enough to shatter a zombie's skull. You just keep walking until you find an open field by a section of river. It's time to start rebuilding society, fertile-crescent style.
But, you think, you're still going to have to move the water from that river to your fields. You may not be able to build a complicated irrigation system, but you can build, or scavenge enough parts, to make Archimedes' Screw. Archimedes' Screw is pretty much what it sounds like. It's a screw. Generally the screw is encased in a tube. The bottom of the screw is placed in the water and the top of it turns on dry land. As the screw is turned, the bottom of it scoops up a little water. The water is caught in the ‘threads' of the screw and winds its way up to the top, where a continuous flow of water is gained without the effort of hauling up heavy buckets.
There are plenty of uses for Archimedes' Screw on a farm. Irrigation is perhaps the most important one, but plant harvesters are horizontal Archimedes' Screws. The twisting motion of the screws pulls up the plants, which the screw then transports to a bin. If you're planning on keeping fish stocks, Archimedes Screws can also pull up small fish without damaging them, allowing you to transport them elsewhere.
Sewage plants also use Archimedes' Screws to drain liquid from full containers (You might want to set aside a special, sewage-only screw for this one. I'm just saying.).
If you're not up to shaping metal or carving wood, a simpler version of the screw is a tube wrapped around a cylinder. It's easy enough for a child to make. Below, see a child making and then demonstrating one, and be shamed.