physics behind the impedance matching performed in this case
I'll see if I can hunt down the complete derivation, but there is part of it on the wikipedia article for quarter wavelength transformers: https://en.wikipedia.org/wiki/Quarter-wave_impedance_transformer .  What it boils down to is that when you have a quarter-wavelength section of transmission line with characteristic impedance \$Z_0\$ terminated in a load of \$Z_L\$, the following relation holds:

$$\frac{Z_{in}}{Z_0} = \frac{Z_0}{Z_L}$$

If you solve that for \$Z_0\$, you get \$Z_0 = \sqrt{Z_{in} Z_L}\$.  If you want to match a particular source impedance, then just set \$Z_{in}\$ to the desired source impedance.  

As for why that is, it has to do with the fact that the line is 1/4 of a wavelength so a reflection off the far end will line up with a reflection from the near end, only inverted due to the 180 degree phase shift.  This has the effect of making the load impedance appear larger or smaller, depending on the ratio of the impedance of the quarter wave line and the load.