If free space impedance is real, why is the electric field not attenuated? Why does vacuum have a nonzero characteristic impedance towards electromagnetic radiation?
Intrinsic impedance given by $\eta=\sqrt{j\omega\mu / (\sigma +j \omega \epsilon)}$. It gives slope of transformation of $\mathbf E$  to $\mathbf H$ and vice versa.
Here $\eta$ is complex.
And in this expression real part is the cause of attenuation and imaginary part is the cause of phase shift.
In case of free space since $\sigma = 0$, we have $\eta = \sqrt{j\omega\mu / j\omega\epsilon} = \sqrt{\mu / \epsilon}$, which is real.
This suggests presence of resistive part in intrinsic impedance which means there should be attenuation. Also curiosity is how free space can offer resistance and however, the expression for electric field in plane wave $\mathbf E = E_0 \exp(wt-\beta z)$ where $\beta =2 {\pi}/{\lambda}   $ and  $\lambda  $: wavelength 
suggests constant electric field. How can we reconcile the real impedance of space with the expression for electric field, which has no attenuation?
 A: It's important to make the following distinction: it's not that vacuum "has" an intrinsic impedance.  It's that electromagnetic waves IN a vacuum have an intrinsic ratio between their electric field (E) and magnetic field (H), which we call impedance.  That impedance is given by Z = E/H, and it is a fundamental constant; it's only when EM waves travel through some medium other than a vacuum that the impedance gets altered.  The units are Ohms because E is measured in Volts/meter and H is measured in Amperes/meter, and 1 Volt/Ampere is defined as an Ohm.  This does not imply that vacuum "resists" electromagnetic waves and dissipates them like a resistor would.
The specific value of Z(in free space) is related to the speed of light, and to the way we define the Volt and the Ampere.  You could think of the "impedance" as being what limits the speed of propagation of the wave, if that is helpful.
A: The so called free space impedance is a fictitious resistance offered by the free space for electromagnetic radiation. It has a meaning when an EM wave passes through free space. Otherwise you cannot measure such a resistance. But for a material, it has an intrinsic electrical or thermal resistivity and it exists all time. But here, the free space impedance doesn't mean such a resistance. If you look for such a resistance in the absence of an electromagnetic wave you cannot find one. Obviously, it is clear from that equation. The impedance is the property of a medium due to the passage of electromagnetic waves through it.  
Now, why the electric field does not attenuate?  
The impedance is caused by the passage of an electromagnetic wave. For the passage of an electromagnetic wave through a wave guide, there should not be an electric field component parallel to the conducting boundary of the wave guide. This creates losses as under such a condition a current is generated on the plates. In free space, there is nothing out there to conduct an hence offers no loss.  
$$E=cB$$  or
$$\frac{E}{H}=\mu_0 c=\sqrt{\frac{\mu_0}{\epsilon_0}}$$  
Permeability is the degree of magnetization that a material obtains in response to an applied magnetic field. Permittivity  is a measure of how an electric field affects, and is affected by, a dielectric medium. It can also be seen as the resistance encountered on forming an electric field in a medium (Source: Wikipedia).  
Hence this resistance exist because there is a speed limit and it is the speed of light in vacuum. One can say that this limit is guaranteed by the permittivity and permeability of the medium. Otherwise vacuum should have offered infinite speed for propagation of electromagnetic waves.
