Getting to know why my meter does what it does: 11 volt LED tester question
It can be more complex actually but you can think it's really the same thing than powering a 3V LED from a 9V battery. You put a resistance in series, to limit current. With 1k resistor, there's 6mA flowing.

So in reality, the open circuit voltage from the meter is up to 12V when nothing is connected, but connecting the probes to a 3V LED will have 3V over the LED, and the rest of the circuit, which may simply be a 2400 ohm resitor inside the multimeter, will have 9V over it, and if it is a 2400 ohm resistor, 3.75 mA would flow.