Relationship of Fourier Transform and Spatial Directivity Pattern
In your second link, the time and space dependence of a simple plane wave is given by

Ψ ( x , t ) = A sin ( ω t ± k x ) 

Just as a Fourier transform of the equation above can be taken from the time domain $t$ to the frequency domain $\omega$, so also can a Fourier transform be taken from the spatial $x$ domain to the wave vector $k$ domain. For the very simple one dimensional plane wave the wave vector reduces to the wavenumber  $k= \frac{2\pi}{\lambda}$ in that first equation.

In general both the position and wave vector are vector quantities and the dependence of a wave on both is given by

$$
e^{j\vec k\cdot\vec r}
$$

The aperture of a microphone array depends on the position of the microphones (a positional quantity), and so the Fourier transform depends on wave vector. 

In Section 1.3.2 of your first link, they are doing exactly that - taking the Fourier transform of the aperture from position to wave vector space, which yields the pattern of the beam in terms of the wave vector.

In the example given in the other answer by @Peter K, think of a plane wave propagating away from the microphone array at an angle $\theta$ .The term inside the exponential, is the dot product of the wave vector and the spacing of the microphone array.