The reason we want the harmonic gauge is that it greatly simplifies the linearized Einstein tensor (I believe it knocks out 3/4 of the terms). So just by inspection you can see (though this is by no means trivial, but we are inspired by E&M), that the greatly simplifying condition is the so-called harmonic, or Lorenz gauge:
$$\partial_{\beta} \tilde{h}^{\alpha \beta} = 0$$
where $\tilde{h}$ is the trace-reversed perturbation in the desired new coordinate system.
The goal is to now show there exists a coordinate displacement $\tilde{x}^μ=x^μ+ξ^μ(x)$ that can produce such a trace reversed perturbation. So if you churn through transforming the metric under the displacement (and that its derivatives are small), you are led to the fact that
$$\tilde{h}_{\alpha \beta} = h_{\alpha\beta} - \xi_{\alpha,\beta} - \xi_{\beta, \alpha}$$
Now if you go ahead and apply this to the definition of the trace reversed perturbation, you find
$$\partial_{\beta}\tilde{h}^{\alpha \beta} = \partial_{\beta} h^{\alpha \beta} - \Box \xi^\alpha$$
So for the harmonic gauge to exist, it must be that $\partial_{\beta} h^{\alpha \beta} = \Box \xi^{\alpha} $. That is the concrete transformation behind the harmonic gauge, and so if you want to come up with concrete examples, this is the equation to try plugging things into.
For example, if you plug in the $h^{\alpha \beta}$ you solve for under the harmonic gauge, you can see what coordinate shifts could produce it.
So intuitively, all you've really done is introduce a wavy coordinate system to describe waves. A great example of why this is true is the fact that neighboring geodesics in linearized gravity representing objects with no 3-velocity will actually show oscillating proper distance to each other.
The oscillation of space itself (the physical metric) from gravitational waves can be understood as the cause of this, or equivalently that the lines of constant coordinate values wave through spacetime.
