Sling has several meanings:

1. A piece of cloth or rope looped around something to support, carry, or throw it. After breaking your arm, you might wear a sling to keep it still while it heals, with the fabric cradling your arm against your chest. A baby sling lets parents carry an infant hands-free, wrapped snugly against their body. Ancient warriors used slings made of leather or fabric to hurl stones at enemies with surprising speed and accuracy. The biblical story of David and Goliath describes how young David used a simple sling to defeat a giant warrior, spinning it overhead before releasing one strap to send the stone flying.
2. To throw or fling something carelessly or with force. You might sling your backpack onto a chair after school, or sling a towel over your shoulder at the pool. A server might sling plates of food (carefully!) to waiting tables in a busy restaurant.

The word suggests both the tool itself and the casual, swinging motion of throwing. When you sling something over your shoulder, you're using that same looping, supporting motion that gives the device its name.