The cause of PLP remains a mystery, but several factors may contribute at once

Factors such as gender, age, location, and level of amputation, and mechanism of amputation (surgical vs. traumatic) show no correlation to incidence rates of PLP. Even though about most phantom pain begins in the first 24 hours after limb removal, it has started decades later for other patients. Usually, phantom pain is not continuous, but rather episodic, occurring for seconds to hours at a time. Sometimes it goes away for good.

Major research fails to address morbidity from PLP, yet one can imagine the debilitating effect of agonizing pain. Mortality directly caused by PLP is difficult to imagine.

Central and/ or peripheral nervous system mechanisms have been proposed for PLP, yet the exact cause still remains a mystery. To (hopefully) keep you from nodding off, I'll limit my discussion of etiology and pathophysiology to the strongest theory, proposed by Ramachandran and Hirstein. This "multifactorial model" cites five contributing factors, taking into account the considerable variation among individual experiences. 

1. Residual limb neuromas: once a nerve is severed, it may sprout branches that lead to increased sensitivity and inappropriate ie. nonprotective input to the brain. This is a peripheral factor.

2. Cortical remapping: magnetoencephalography has provided evidence that "cortical areas representing the amputated extremity are taken over by neighboring representational zones in both the primary somatosensory cortex and motor cortex." (Weeks et. al, 2010, 279) Since this is one of the most accepted theories behind PLP, and it requires a little background, I've placed a fabulous Khan academy video below that gives a little background into the somatosensory cortex (which is conceptually similar to the motor cortex).

Khan Academy: how the somatosensory cortex works

https://www.youtube.com/watch?v=3jf2l9ma6SM

I have to confess, I haven't been able to find a source that explains exactly why cortical remapping causes pain. 

I have to confess that I don't understand quite why cortical remapping causes pain, and I have been able to find a source with a clear explanation, so I hope to get back to this. Note that this is a central problem. Below I've included a picture illustrating cortical remapping, which explains the what but not the why.

How cortical remapping

http://www.google.com/patents/US20110065505

3. Monitoring of corollary discharge from motor commands to the limb. It's explained like this: "Rather than the brain receiving information that the limb is immobile, it fails to receive feedback from a newly amputated limb to confirm that a motor command has been followed." (Weeks et. al., 2010, 280) Another central problem.

4. One's own body image. This is the most self-explanatory central problem.

5. "Vivid somatic memories of painful sensations or posture of the original limb being 'carried' over into the phantom." (Weeks et. al, 2010, 280)

EDIT: 

This TED Talk by Dr. Ramachandran added clarity to the idea of cortical remapping. 

https://www.youtube.com/watch?v=l80zgw07W4Y

Sources:

Weeks, S. R., Anderson-Barnes, V. C., & Tsao, J. W. (2010, September). Phantom Limb Pain Theories and Therapies. The Neurologist, 16(5), 277-286.

Canavero, S. (1994). Dynamic Reverberation. A Unified Mechanism for Central and Phantom Pain. Medical Hypotheses, 42, 203-207.