How was the polarization experimentally measured in the BICEP2 experiments and why did they look specifically at B-modes? Why is it implying the existence of gravitational waves and the need to quantize gravity? Moreover, why is it implying inflation as a unique option?
I can answer partially to your questions.
How do we measure polarisation ?
They measured the so-called $Q$ and $U$ Stokes parameters. There are four Stokes parameters : $I$, $Q$, $U$, and $V$. $I$ is the intensity, that we already know a lot about (temperature), then $Q$ and $U$ are linear polarization along axis that are tilted with an angle of 45° with respect to each other, and V describe the circular polarization. There is no physical phenomena to create circular polarisation in the CMB, that is why we ignore it and only consider $Q$ and $U$ (maybe $V$ is still measured for calibration or whatever but it is near zero). Wikipedia
Why do we look specifically at $B$ modes ?
The $Q$ and $U$ parameters can be easy to measure experimentally, but are not very handy on a theoretical point of vue since they depend on the system of coordinates that you use. What is more interesting is the $E$ and $B$ modes of the polarisation. Given a $Q$ and $U$ map, you can decompose the polarisation into $E$ and $B$ modes (it can be a tricky problem however when you have a lot of pixels or some borders).
The $E$ modes are often referred as the "gradient" part of the polarisation, while the $B$ mode are the "curl". The thing is that $B$ modes can only be created with tensorial perturbation (i.e. gravitational waves), and not scalar perturbation (temperature).
Last, I don't think observing $B$ modes has for only implication that the inflation is the only possible theory. You can always find other fancy theories that would produce the same signal. But those are generally more complicated and less convincing in general. This is why people consider that observing $B$ modes is a solid hint for inflation.
Here is the link to the paper.
This is a detector at the south pole with a telescope focusing the cosmic background radiation (CMB), on a focal plane equipped with incident energy sensitive detectors, the signal going to a data acquisition system. All the system is kept at 4 kelvin because the incident CMB is very very low frequency: a black body radiation system of temperature 2.7 kelvin.
It seems that the detectors in the focal plane are paired
. . .
In section 4.1
This says that it is the relative difference between adjacent temperature detectors that will give the information on polarization. These transition edge detectors use superconductivity to attain the sensitivity to polarization.
This is exciting because it is a first detection of an imprint of the graviton. To have gravitons, gravity must be quantized. What the result tells us is that the approximate field theoretical models used for the first instances after the Big Bang, which assume quantization of gravity are on the right path, which means gravity has to be quantized.
This observation excludes models that do not have this high intensity gravitational fields that the Big Bang standard cosmology has. I suppose theorists with different theories will be scrambling to fit the data.
In this blog post a good discussion of the results is given by prof. Liam McAllister.