This is an unnecessary special effect that occurs when players instantaneously absorbs far more damage than is necessary to kill them (read: excessive). Because Quake is such a realistic physics engine and knows that the energy of a system much be accounted for at all times, it channels all the additional damage energy into blowing the body into a variety of meaty chunks.
These are gibs.
Both a noun and a verb - to blow someone into gibs, there are a variety of spinoffs: gibbage, gibb, gibber, etc. Pronounced jib, and does indeed originate from giblet - pronunciation and origins are confirmed by id. Originally this came from Doom (although I believe you could gib in Wolfenstein this wasn't yet a term).
Keep in mind that the following information is Quake pertinent only: Gibs come in packs of 3, with a head tossed in as well. While flying through the air they spray blood everywhere.
Here's the science behind it!
The victim, upon being killed, must have less than -40 health. ie, if you have a rocket launcher and you score a direct hit on someone with 50 health, you do 120 damage taking their health to -70. They have been gibbed. The total damage applied must occur simultaneously. In the original Quake you cannot kill someone and then axe, rocket or shaft their intact body to turn it into chunks (this 'oversight' was corrected in some custom mods as well as Quake 2 and Quake 3)
For even more spectacular effects, gibs don't just drop to the floor. If you've done enough damage you can blow someone to all n corners of the room. The speed (if any) of a paritcular gib is calculated thus:
x/y components of velocity are randomly calculated between -100 and 100, while the z-component is calculated betweeen 200 to 300. They are then multiplied by some factor which depends on the health:
-40 to -49 health - 0.7, usually a super shotgun blast to a low health person, who turns into a meaty pile.
-50 to -199 health - 2, your most common rocket launcher gib, and occasional Quad Damage (super) shotgun/super nailgun frag, making a healthy meaty red splatter.
-200 onwards health - 10 (yes, that's 10). What can I say? Quad rockets hurt. Get out the carpet cleaner doris we got some cleanup to do.
Part of the Quake metanode