Ohm's Law:
E-IR
I=E/R
R=E/I
If you take the voltage reading across a resistance (in this case the internal resistance of the battery) you will read the voltage across the six battery cells (~2.1vdc per cell). By connecting the remainder of the starter circuit when you activate the starter switch you will read a voltage which is now split with the circuit's other resistance drops across the wiring, starter motor coils, and any other circuit that remains active during the start cycle. If there is insufficient current flow due to excess resistance like corroded or loose connections, bad wiring, sulfated battery cell(s), etc., then you will not be able to start the engine as well as not enough current/voltage to illuminate/activate any lights or additional circuits, etc.. The voltage test John and David suggested makes it a quick and easy test if you don't have a load tester or carbon pile.
The one bad battery may draw the voltage/current down enough to cause issues, even it the other battery is good. I have not tried to start with only one battery, but have heard of others just using one to start their vehicles. Maybe you could remove the low reading battery as a test to see if the good one makes a "little stronger" attempt to start the engine. At 12.1 vdc that battery is only ~50% charged.