According to Halo.wikia.com (sourced from the Halo encyclopedia):
When activated, the Halo rings would wipe out all sentient life within three radii of the Milky Way's Galactic center, by sending out radiation, targeting certain cells in the nervous system, which includes, but is not limited to, neurons, by harmonizing all neurological frequencies.
The pulse targets the nervous system of sentient life forms via the issuing of a harmonic frequency
Presumably this operates similar to how an EMP destroys electronics - the burst of radiation destroys necessary components for operation of electronics. If all the neurons are destroyed, brain function would instantly cease, killing the organism. Presumably this means that sentient species as varied as Humans, Kig-yar, and Jiralhanae share some similar brain structures.
Edited, based on new information from HALO: Silentium (Spoilers)
The Halos also target anything, not just biological, that uses neurons. The Precursors' technology was based upon the unknown capabilities of neural physics. When the Halos fired, all Precursor architecture was destroyed as well. Presumably since the Precursors are supposed to have created life in the galaxy, they are the reason that neurons seem to be common between sentient species.