How to protect LEDs from high voltage during NPN transitor transitions?
There's no problem with the circuit. 

The voltage across the LEDs take a bit of time to build up to the 3V because of capacitance the LEDs have. That's all you are seeing. 

When the transistor turns on, the VLED voltage drops to about 0V quickly, then over a period of time the capacitance charges and the VLED voltage rises to a voltage such that the LEDs are conducting essentially all the current through R1. 

Try plotting the actual voltage across the LEDs. If you are using LTspice, then just edit "V(vled)" to "V(vled) - V(vgnd) by right-clicking on the label and changing it in the Expression Editor. Also plot the LED current. You won't see excessive forward or negative voltage, nor will you see excessive LED current.