There is only one rudder on the aircraft, so speaking of "the standby rudder" would be incorrect -- to describe it that way suggests the presence of a "main" rudder and a second "standby" rudder, and that's not what's on the 737. What you do have is the standby hydraulic system, which is used to provide backup hydraulic power to a few components (listed in @Kevin's answer) when one of the main hydraulic systems (A or B) becomes inoperative.
So the better way to think of "Stby Rudder" would be "Standby Hydraulic System powering the rudder" -- and on the switch, "in lieu of Hydraulic System A" or "... B". Those switches are used either when a hydraulic system has failed, or to test the standby hydraulic system. The three positions of each switch are "__ ON" -- so the respective (A or B) hydraulic system is powering all flight controls, "OFF" -- so the respective hydraulic system is not powering any flight controls, and "STBY RUDDER" so that in addition to the "Off" functionality, the standby hydraulic system is activated & powering the rudder. (Why just the rudder? The ailerons & elevator can be adequately controlled in Manual Reversion -- zero hydraulics -- mode, but the rudder cannot. It has to have hydraulic power to provide the deflection required in case of an engine failure.)
The standby pump comes on automatically under various conditions, essentially all related to loss of a main hydraulic system. So if a main system were to fail on short final, let's say, you'll have the Standby pump running to power the Thrust Reverser on the side where the failed system would normally power it -- no need for an immediate reaction to put the flight controls to Standby Rudder during a critical phase of flight. In that scenario, your good hydraulic system continues to power all flight controls just fine.
As @Koyovis describes, the RSEP modification added some additional safety features and the "Stby Rudder" indicator light. That light is new with the RSEP mod; the switch position had always been labeled like this. Essentially, the RSEP mod added sensors to detect if the A-system and B-system rudder PCU's were opposing each other. This is driven by the theory that a rudder PCU can "run away" and cause uncommanded actuation of the rudder to a hard-over position. Were this to occur, the RSEP mod enables the Standby Rudder PCU (i.e. the PCU for the rudder that is powered by the Standby Hydraulic System) to become active & "break the tie" so that whichever PCU is responding to flight control inputs "wins" and overpowers the runaway PCU. The Stby Rudder On light will come on if the Standby Rudder PCU is powered, and if the flight crew didn't command this by putting either Flight Control switch to "STBY RUDDER" then they know it is happening because of a "force fight" between the A and the B PCU's and that the Standby PCU is working to keep things under control. I've not heard of that happening during actual operations; maybe Boeing has.