@Malcolm gave an expert explanation of the dealiasing of the FFT based convolution sums. My intuition tells me however that this is still not the complete answer @abberation needed, so I'll give few more sentences that reflect my own struggle for understanding of numerical methods.
Dealiasing doesn't act as a numerical dissipation, but it does the same thing as numerical dissipation - it prevents your simulation to blow-up.
How does your simulation blow-up?
There is an unphysical transfer of energy from high to low wavenumbers which causes, first the instability (often followed by a simptome - 2h waves, h-grid node distance, the smallest waves that can be represented on a grid).
The reason behind this is aliasing of high wave numbers to those low wavenumbers that can be represented by the grid. This causes "spectral blocking" - the unphysical increase of the amplitudes (read energy) of the high wavenumbers, which gets worse as you approach the upper limit of the represented wavenumbers.
First remedy (Philips, 1959): Filter out, or make equal to zero, upper half of the represented wavenumber spectrum.
Second remedy (Orszag, 1971): The remedy known as the Orszag's Two-Thirds rule. Steven said "filtering out half of the wavenumbers is a waste, we should keep 2/3N and filter out only the upper 1/3N."
Our simulations sometimes tend to blow-up. If you live in second order, finite volume universe (like the author of these lines did), and know nothing about Spectral Methods, they tell you "Oh your simulation is blowing up? Try using first order upwind instead of central scheme, or blend your central scheme with 10% of upwind in deffered correction approach. That will add dissipation but will prevent your simulation from blowing-up! Oh you're doing LES? Then you shouldn't use upwind, unless you are performing ILES, use just central scheme and Smagorinsky will model the physical dissipation that happens on lowest scales."
If you live in the Spectral Methods universe, you learn how unphysical transfer of energy takes place, and you find the remedy in dealiasing by Two-Thirds Rule or "padding".
My inspiration for this post is a wonderfull book by J. Boyd "Chebyshev and Fourier Spectral Methods", section 11.
My advice is:
Act physically or spectrally, but think spectrally!
(To paraphrase well known "Think globally, act locally.")