Why do they leave unattended hockey sticks on the ice?

Answer 1

A broken stick is left on the ice because it's a penalty to use it, and a dropped, but still in-tact stick is sometimes left on the ice because picking it up will put its owner out of position so they wait for a good opportunity to grab it.

Answer 2

If a stick is broken it must be dropped immediately or the offending player is charged with a minor penalty. (This is to reduce or eliminate any potential injuries occurring from being hit, poked, stabbed or cut by the broken stick.)

The goaltender is the only player allowed to player with a broken stick.

In other circumstances, a player may find it easier to grab a fresh stick, or make a line change instead of spending time retrieving the dropped stick.