Why is the time constant of RL series circuit be \$\frac{L}{R}\$?
simulate this circuit &ndash; Schematic created using CircuitLab

The transfer function of the simple L-R lowpass is 

H(jw)=Vout/Vin=R1/(R1+jwL1)=1/(1+jwL1/R1)

Now - the factor for the frequency variable L1/R1 is DEFINED as the time constant (and it has the unit "second"). 

From system theory we know that the step response in the time domain contains an exponential expression exp(-t*R1/L1)=exp(-t/T1).

Now, you can see that after the time T1=L1/R1 the exponent is "-1". This case DEFINES the time constant: 
The time constant gives the time which is necessary to bring the step response down to the factor 1/e.