There was no explosion. Let me repeat: there was no explosion. What happened was that at approximately 58 seconds into flight, hot high pressure gas from the starboard Solid Rocket Booster started blowing past the O-ring seal in the joint between the aft-most segment of the SRB and the one forward of it. (The SRB is fabricated in segments and integrated via field joints at the vehicle integration facility due to the problem of transporting such a large solid-fuel booster.) By T+59.262 there was a definitely plume emitting from the joint and being directed via aerodynamic forces directly toward the skin of the Eternal Tank.
At T+64.660 the color of the plume changed, indicating burn-through on the ETs outer skin and inner LH2 tank, leaking hydrogen fuel. At this point, the STS was approaching Max Q (maximum areodynamic loads) and was also experiencing large wind shear, which contributed to bending forces on the SRB causing joint seperation. The heat from the plume, integrity failure of the ET, and aeroloading resulting in a failure of the lower SRB strut at T+72.20, allowing the starboard SRB to start rocking around. AAt T+73.124, the upper dome of the LH2 tank began to seperate and leak, and at about the same time the loose SRB contacted the intertank support structure between the upper LOX and lower LH2 tanks, resulting in complete structural failure of the ET at T+73.137. This resulted in spontaneous combustion of the escaping fuel and an increase in net thrust from the lost propellent mass. The heat and stress also caused a leak in the Reaction Control System, causing its MMH fuel to burn hypergolically, although this did not contribute significantly to thrust or damage.
The resulting increase in net thrust, aerodynamic loads, and dynamic instability caused multiple catastrophic failures in the structure of the Orbiter (many components of which had very low structural margins in nominal flight to begin with) which emerged from the fireball. The SRBs, freed from the decaying ET bracing, continued in unguided flight for another ~35 seconds before being flight terminated (i.e. a linear shaped charge blew the boosters apart down a line parallel to their axis) by the Range Safety Officer. The main fuselage of the Orbiter, including one wing, most of the main cargo bay, and trailing umbilicals, went into a more-or-less flat spin that would have prevented the crew from egressing even if they were conscious at this point. The Orbiter Main Cabin impacted the Atlantic Ocean approximately 2 minutes, 45 seconds after breakup. Some of the Personal Egress Air Packs were later found to have been actuated, leading some to the conclusion that despite the tremendous G forces just prior and during breakup at least some crew remained alive until impact, but this is inconclusive. Regardless, the crew would have been killed by the unbraced impact, and what remained of the Orbiter sank into about 90 feet of water off the Florida coast.
While the spontaneous combustion due to leakage of the ET was visually impressive, the damage it did to the Orbiter was minor, probably no more than charring insulation. The real damage was done by a combination of aerodynamic loads on the destabilized Orbiter and the sudden impulse from the increase in net thrust from the loss of propellent mass. Even if the Orbiter had survived the sudden stress of this event, there is no conceviable way it would have been able to perform a successful intact abort manuever; the earliest possible time for the first abort mode, called Return-To-Launch-Site (RTLS) called for initiation of manuever ops at approximately T+140 seconds and >200k ft of altitude; the Challenger breakup occured at roughly 65k ft, far too early. RTLS also requires use of both the SRBs (although they are staged earlier than the planned mission sequence) and the Shuttle Main Engines. Catastrophic failure of SRBs during ascent was then and is now considered an essentially unrecoverable failure. (Whether the STS could dead stick an RTLS maneuver in the case of complete SME failure or loss of the ET post SRB staging is highly questionable at best; I believe current plans in such a case call for a manual bailout and abandonment of the Orbiter.)
The problems with the O-rings, and general consensus on likelyhood of failure was understood long before STS-51-L came apart like a cheap gold watch. From Space Shuttle: The History of the National Space Transportation System: The First 100 Missions by Dennis R. Jenkins (which is where I’m drawing the bulk of this information), pg 281 (citing the report of the Presidential Challenger Accident Review Board): *O-ring anomolies had been detected to varying degrees on 12 previous flights. Erosion of either the primary or secondary O-rings had been seen on Flights 2, 10, 11, 12, 15, 16, 17, 18, 20, 22, 23,. and 24. A more serious problem, the actual blow-by of exhaust gases past an O-ring had occured on Flights 11, 12, 15, 16, 17, 22, 23, 24. All of these anomolies were recorded upon occureence, and this data was known at the time of the Flight Readiness Review for STS-33/51-L. Failure analysis conducted as early as 1979 on the Space Shuttle had concluded that one in fifty flights would encounter a catastophic accident during ascent, and one in 100 would fail to land successfully. The failure analysis have been updated repeatedly, and until recently, had always reached much the same conclusion.*So, anomolies and potential for failure (the numbers for which have essentially been borne out by later experince) was known. There has been the suggestion–echoed in the Wikipedia article–that the dramatic wind shear was primarily responsible for the failure of the SRB, but while wind shear may have definitely contributed to joint seperation and the speed of the breakup, problems with the joint had been observed and analyzed long before. Erosion of the two O-rings in the joint, after initial observations, were accepted as nominal operational behavior, even though any erosion of the O-ring compromised joint integrity outside of design specifications. Furthermore, the design of the joint was such that a partial failure would tend to leave the joint even more exposed and prone to opening. Worse yet, the way in which the O-ring groove was designed put the O-ring into an out of spec (or at least edge of spec) condition, compounded by the famously demonstrated lack of resiliency of the O-ring material at low temperatures. Following the Challenger disaster, the SRB and in particular the joint was redesigned in a way that essentially inverted the joint, making it more prone to self-sealing in the case that bending forces pried at the joint, and adding an intermediate O-ring as further backup.
While the use of O-rings, and in general segmented booster with field joints is still somewhat questionable, the fact is that in practical terms the SRB has a fantastic record of success for a solid booster with only one mission failure in over 230 operational launches, not even counting test launches and static fire testing, and is planned as the first stage for the Ares I booster for the Contellation program, and remains the only solid propellent booster man-rated for orbital flight.
I highly recommend the above-referenced book as an almost comprehensive source of information regarding the history and development of the Space Transportation System (U.S. Space Shuttle), and I wish somebody would do something like for the Soviet Buran program.