Why a higher compressor pressure ratio results in higher Thrust and lower TSFC?
Because the constant pressure lines are diverging with increasing enthalpy.

Have a look at the right most diagram.
The two lines are constant pressure lines, with the higher line indicating the higher pressure, and as you can see, they are diverging.

1 is the beginning of the cycle, thus at ambient temperature

From 1 to 2, we increase the temperature, and go to a higher pressure in the compressor. This will cost some energy, which is delivered by the turbine.

From 2 to 3 we increase the Temperature at the the same pressure by adding energy.

From 3 to 4, we decrease the temperature and pressure in the turbine (and use some of the extracted energy to drive the turbine)

From 4 to 1, we use the energy in the flow to propel us, and we go back to ambient temperature.

In principle we could also skip the higher pressure, and simply go from 1 to 4 and back to 1 (which is just heating air, and propelling yourself by the heat.)

But by moving to a higher pressure, we can take advantage of the divergent pressure lines. This yields us a benefit because the distance 3-4 is larger than 2-1. This means that if we use the turbine to drive the compressor, we gain some 'free' energy.



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Following OSUZorba's comment:

To illustrate the point of the 'free energy', look at the image below, here two extra stations are added the inlet, which isn't that important for now, and the nozzle. Note that the vertical distances of the compressor and the turbine are equal (they have to be, because one drives the other), thus 
$$ \begin{aligned}P_{comp}  = P_{turb}  &amp;\Rightarrow \Delta T_{comp} = \Delta T_{turb} \\ &amp;\Rightarrow (T_3 - T_2) = (T_4 - T_5) \end{aligned}$$

But because of the divergent nature of the lines, we have some extra energy left. We use the nozzle to optimally expand the high pressure, high temperature flow and extract the 'extra' energy.



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