Why the voltage of supply got stuck when connected with leds?
Your power supply can limit current - see the "coarse" and "fine" control knobs on the right side under "Current."  Presumably, you had the current limit set to around 190mA.



As Wouter van Ooijen says, this also provides a safe, simple method of getting the data for your V-I chart.  


Set the voltage at a safe maximum (from the LED datasheet) 
Set the current down to 0. 
Read voltage, 
Increase current, 
Read voltage 
Repeat steps 3 to 5 until the voltage reaches the maximum you set or your current reaches the maximum for your LED. 
Finished.




It looks like you are measuring 1 LED from each module, with the two LEDs in parallel.  You realize, that your readings may also be influenced by whatever else is in the circuit?  That's the other LEDs, the resistors, and what looks like it might be a regulator or controller of some kind?