What is VBEn and why did it change?
Base and emitter together forms a pn-junction. And hence the current through it varies exponentially with the voltage across it. 
$$I=I_0\exp(v/v_T)-I_0$$

If you increase the voltage across this junction from 0V, the current will be negligible till some voltage. After that you can observe sudden increase in current. This voltage is what you are referring to as "turn-on voltage". This is not actually a constant. 

The voltage across the junction will not vary much after turn-on voltage (only \$\approx 60\,mV\$ change can be seen for 1 decade change in current) because of the logarithmic variation of voltage with current. That's why people assumes it as a constant. 

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