Design of attenuators in microwave systems
Such microwave attenuators are very common, and work by absorbing the Microwave energy into a lossy material which then dissipates this energy as heat. 

However, as you might have learnt from EM theory, introducing any material discontinuity causes reflections because of the impedance mismatch at the discontinuity. This means not all the energy is absorbed by this lossy material thereby reducing its effectiveness as an attenuator. 

The taper here is performing the role of an broadband impedance matching section. You might be familiar with the Quarter Wave Transformer, i.e a section of line that performs narrow band impedance matching. Now intuitively, if you chain a series of such sections you can expect a wider bandwidth of the impedance match. The taper is simply an extreme example of this where the impedance matching sections is essentially just a section of line where the impedance varies continuously.

Interestingly, the shape of the taper defines the frequency response of the Input Reflection Coefficient! So this shape is not arbitrary, and is rather carefully chosen to achieve a certain frequency variation of the Input Reflection Coefficient (looking into the start of the taper). 

Do you see the strong similarity between this exercise and that of the more familiar filter synthesis where we choose filter coefficients (hence shape of impulse response!) to attain a certain frequency response such as Butterworth, Chebyshev etc?. By analogy all of this also applies to taper synthesis where we instead shape the taper to attain a certain shape of \$\Gamma_I(\omega)\$! 

If you would like to know more look for:
1) Theory of Small Reflections
2) Tapered Transmission Lines 

If you are instead interested in the maths behind this, leave a comment and I shall try and put together a quick derivation as an edit :)