The missile traversed the airframe from port to starboard, travelling immediately beneath the passenger deck, and almost parallel to it, yet with a very, very slight up-angle, and just ahead of the bulkhead that is the forward wall for the void that is ahead of the main fuel tank.
The missile did not explode in the aircraft.
The missile disrupted the integrity of the deck/bulkhead structure. The missile's accompanying supersonic shock wave's pressure, is a contributing factor that is often overlooked; the missile did not have to explode, though there is evidence that it did, after exiting the aircraft.
TWA 800 wandered into an anti-missile test; the U.S. Navy, the other branches of the military, did not shoot down the aircraft.
"Inquiring minds want to know."
I don't think it takes a genius to surmise that what happened to Flight 800 was a major fuel leak, with engine exhaust as the ignition source. The "streaking light" seen by so many people from the ground was probably a series of ignitions prior to the main explosion, giving the impression that something was rising up to meet the aircraft.
Knowing very little about either aircraft or missiles, I lurked about for quite some time on one of the more popular aviation disaster newsgroups to try and get a feel for the technical issues involved in the discussion of Flight 800. One particular piece of information I garnered from experts who posted there is that there is no known missile whose engines would have continued to burn at the distance necessary to have fit the various missile scenarios described by the conspiracy theorists. Another is that the launch of such a missile would literally have lit up the horizon for many miles around; something not seen by any of the witnesses.
13,800 feet is just in the outside range of a shoulder-fired missile--but that's assuming that you're firing straight up at a stationary target. If the target is downrange even by a little bit, and moving, you've got to factor in the distance downrange as well as its rate of speed, into the equation.