Why should dynamical downscaling of meteorological data be made incrementally? When downscaling meteorological data by using limited area models, many support forums recommended you to downscale incrementally in small steps. For example let's assume that your input data is on a 27x27 km resolution and you need data on 1x1 km resolution.

A suitable setup would be to downscale in three steps, incrementally increasing the resolution:
27 km -> 9 km -> 3 km -> 1 km

Why should I not simply use the following set-up?
27 km -> 1 km

The main motivations I've seen for this is that the intermediate domains (9 &amp; 3 in the example) won't require that much computational resources compared to the final domain (1 km). But that can't be the only reason.
The basic answer to your question is that there needs to be an overlap between the spectra of the two domains. One way to think about it is that there needs to be a connection between the Kolmogorov energy cascades (https://en.wikipedia.org/wiki/Energy_cascade) of the two domains. 

Kolmogorov energy cascade (D. Newman, U. Alaska Fairbanks).

Each domain is going to be forced (have energy input) at a relatively large scale and it is going to assume that the energy that it can't resolve at sub-grid scales is going to be dissipated. If there is no (or little) overlap between the two grid resolutions, the energy from the large domain (parent) will be dissipated by viscosity at sub-grid scales that the small domain (child) cannot resolve and there is not going to be an appropriate input of energy from the parent to the child. 

The reason for the odd numbers in the ratio is given in Why is WRF most often configured at 3:1 nesting ratio?