Let me first point out that beating occurs when the difference in frequencies (time frequencies or spatial frequencies) is small, compared to their magnitude: $$|k_1-k_2|\ll k_1, k_2, \frac{k_1+k_2}{2}.$$
Apart form them the reason is that the two waves have nearly the same frequency/wavenumber, but the phase shift between these two waves is very slowly changing, so that from being in-phase, they gradually become out of phase, that is from constructing interference one gradually moves to destructive interference and then back (image source):
Taking expression from Landau&Livshits, some things are easier to derive oneself than to understand a derivation:
$$
\sin(k_1x-\omega t)+\sin(k_2x-\omega t)=\\
\sin\left(\frac{k_1+k_2}{2}x+\frac{k_1-k_2}{2}x-\omega t\right)+\sin\left(\frac{k_1+k_2}{2}x-\frac{k_1-k_2}{2}x-\omega t\right)=\\
\sin\left[k_{mean}x-\omega t+\phi(x)\right]+\sin\left[k_{mean}x-\omega t-\phi(x)\right]=\\
2\sin(k_{mean}x-\omega t)\cos[\phi(x)]
$$
If the phase difference were constant, we would have fully constructive interference for $\phi=\pi_n$ and fully destructive interference for $\phi=\frac{\pi}{2}+\pi n$.
Since both waves have different wave number, beats can exist. Furthermore, they remain stationary for some reason. I unable to intuitively understand why this happens. Any help would be appreciated.
The beats are "stationary" in space because the difference is between wave numbers rather than between frequencies, so the beating occurs in space, rather than in time - i.e., as we change our position, rather than as time flows (at the same position.)
What might make this case seem somewhat unusual is that it implies that we can create waves with arbitrary pairs of frequencies and wave numbers, which is typically not the case, as frequency and wave number are related via dispersion relation (e.g., $\omega=ck$ for light or sound), which is encoded in the wave equation.^1 That is usually beating would occur in both time and space. That is the two waves in the example discussed are not solutions of the same wave equation, and thus represent a somewhat exotic example.
^1 If
$$
\frac{\partial^2 u(x,t)}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2 u(x,t)}{\partial t^2}=0$$
then by plugging a solution in terms of a plane wave
$$
u(x,t)=e^{ikx-i\omega t}
$$
we readily get
$$
-k^2+\frac{\omega^2}{c^2}=0\Rightarrow \omega=\pm ck
$$
Thus, the only wave to have two different wave vectors corresponding to the same frequency is when they have different directions. In one dimension this means $k_2=-k_1$, so that $k_1+k_2=0$. The condition for beating stated in the beginning is manifestly not satisfied.
