Not all processors work the same way. Probably the most-imitated one is the Aphex Aural Exciter, which was the original.
The quotes below are from http://www.soundonsound.com/sos/1995_articles/feb95/exciters.html
The Aphex Aural Exciter (the first one):
...some of the input signal is diverted, via a side-chain and a high-pass filter, into a harmonics-generating circuit. The high-pass filter is necessary to remove unwanted low frequencies which, after processing, might result in a muddy or discordant sound. The filtered signal is then processed dynamically to add phase shift and to create synthesised harmonics which are musically related to the original signal. A small amount of this signal is then added into the output, which has the effect of reinforcing and emphasising transient detail without significantly increasing the signal level.
The BBE Sonic Maximizer:
BBE Sonic Maximizer works not by adding harmonics, but by introducing phase changes and dynamic equalisation, which just redistribute those harmonics already present...
...This particular system divides the audio spectrum into three bands before applying corresponding frequency-dependent delays, low frequencies being delayed the most and high frequencies the least. This arrangement creates opposing phase delays to those encountered during passage through normal analogue signal processing chains as well as in conventional loudspeaker reproduction. The aim of these compensating frequency-dependent delays is to mirror them and thus cancel out this particular source of sonic distortion.
The SPL Vitalizer:
The Vitalizer works by first generating a side-chain signal from the main signal; the frequency response of this side-chain signal is then modified both additively and subtractively. Because of the way filters interact, the impression of an increase in both bass and brightness is created when the side-chain signal is added back to the original, while the mid-range is brought into sharper focus, increasing the sense of transparency.
The Dolby algorithm (used in several products?):
Rather than adding harmonics or using simple dynamic filters, the Dolby approach relies upon treating a side-chain signal via a bank of complex filters, which modify their characteristics according to the nature of the input signal. The filtered signal then appears to be heavily compressed before being added back into the main signal path.
Many systems offer some provision for enhancing the low-frequency spectrum, most commonly by further dynamic equalisation which raises the relative amplitude of any low-frequency content as the overall volume drops. In concert with the HF boost and mid-range depression, this LF boost results in the classic 'smiley' equalisation curve -- a dynamic 'Loudness Filter' if you like -- which is widely recognised as producing a pleasing and appealing sound characteristic...
...This is achieved, in part at least, by altering the relative phase at which the dynamically equalised low-frequency region is added back into the main body of the sound.
The overall psychoacoustic theory:
The answer is that whenever an audio signal is subjected to distortion, intentional or otherwise, high-frequency harmonics are produced. Normally, these sound pretty unpleasant, as they are not always musically related to the original sound, but by using filters to confine the distortion to a specific part of the audio spectrum, it is possible to create the illusion of additional high-frequency detail without musical dissonance.
If an electronic processor is able to delay the low frequencies slightly by means of deliberately introduced phase shifts, it is possible to restore the original phase relationship, making the sound source appear to be closer. This is why processed sounds seem to be very 'up front'.