The humble slingshot is an amazing tool before you learn the engram for bows and arrows in Ark Survival Evolved. The slingshot is good at knocking out low tier animals on the Ark which can then be tamedTo use a slingshot, and bow for that matter, you need to pull back the rubber band and then let it loose. The amount you pull back affects how much damage the stone will do. It will take you longer to hold down to full power, but it is the best way to deal the most damage per second for the slingshot. If you hold back too long the slingshot becomes unsteady and it is more difficult to be accurate with it.
While the slingshot is easy to craft and repair, it does break with little use, so it is best to keep at least two in your hot bar when taming something like a triceratops. Always pick low-level creatures when taming with a slingshot or club. High-level ones seem to die more often, and it will take a lot of stones, and slingshots. The added weight makes it hard to escape the wild creatures or chase them down when they torpor run.
The slingshot is best when used in combination with a bola. The bola will hold parasaurs, pteranodons, and raptors while you knock them out. Obviously, you want a very low-level raptor if you are at a level when you are stuck with a slingshot. If slingshots did not disappear after you dropped them from your inventory the beaches of old Ark servers would be littered with them like beachgoers on a holiday weekend.