A stethoscope can be bought from Martin Thwait at the Rogues' Den. Level 50 Thieving and Agility (can't be boosted) is required to access Martin Thwait's store and purchase the item. You can also obtain the Stethoscope without 50 Agility by having 99 Thieving.
A stethoscope increases the chance of successfully cracking the Wall safes in the centre of the Rogue's Den main chamber. It has no effect on any other thieving activity. Note that you do not need to "Use" the stethoscope on the safes, it just has to be in your inventory.