White LED lantern powered from USB port. Jaroslav Belza
R1 Limits the LED current when T1 is fully Turned on or saturated. P1 forms a Variable Potential Divider, The Turning of P1 Pot gives a 0 – 5 V Variable Voltage Proportional to the turning, if it is a Linear Pot.(Audio Systems use a Log Pot as the Ear Hearing response to sound is Non-Linear.). As the impedance of this Voltage source is a bit high, it is from a 5K pot, it cannot be loaded with a low value resistor like 100E or 1K.
The R1 should limit the current to an optimum value in relation to LED Brightness vs. Battery Life. T1 chosen must have a Max Ic of at least Five times the LED current, for a long lasting product.
R2 you can figure out from the beta of the transistor. That you can learn at Transistor Operation – HyperPhysics.