Electronic – the correct supply voltage for LEDs

First, this is not a good circuit. The current with the switch open depends on the current amplification factor of the transistor, which can vary widely. If you just want to vary the brightness of the LEDs you could put an extra series resistor (beyond R2/R2, value to taste) and short that resistor with your switch (change R2/R3 to limit the current to your desired value).

The reason that you don't burn out your LEDs is probably that you used a 9V battery, which has considerable internal resistance. In your simulation you probably used an ideal voltage source, which has 0 internal resistance.

Another reason might be that a LED does not die immediately from a high current, it merely has a significantly shorter lifetime.

The schematics looks fine, and MCP6002 should be able to drive LED without any problem (it has I_SC of 23mA and V_OL of 25mV). Most likely the circuit is mis-assembled. You need to troubleshoot it:

1. Check opamp output. It should be <2 volts, ideally <0.1 volts.
2. Measure your power supply to make sure it is really giving you 5 volts.
3. Check voltage on the connection between resistor and LED. It should be ~1.5-2 volts. More than that means LED is broken or backwards or opamp is bad. Less than that means resistor is of incorrect value, or power supply is bad.
4. Check voltage across LED. It should be ~1.5-2 volts. If it is close to 5 volts, the LED is broken or installed backwards. If you have problems here, but steps 1-3 check out fine, you have power bus oscillations in your system.