I've answered the question of steel failure and structural failure before, so here's the condensed version:
The jet fuel burned off quickly, but not before spreading through at least the affected floors in a matter of seconds. Because of the interior structure this fuel was distributed unevenly. According to the head of the materials engineering department at MIT, a temperature difference of only a couple hundred degrees from one side of each support beam to the other would cause it to warp, not soften or melt. The fire wouldn't have had to burn at more than a few hundred degrees to bring down the buildings. This would be enough to put significant force on the affected floors. Each floor was attached to the load-bearing outer cage with angle brackets; the force from the warping support beams started to rip the brackets out of the cage. As the brackets popped, the load was transferred to the other brackets, already stressed from the unusual motion of the building. Once enough brackets failed, the floor would break free and slam down onto the floor beneath it, compromising the brackets attaching that floor. The compression of the air between floors blew out windows and ejected debris, causing the puffs of smoke that can be mistaken for demolition charges.
Because the outer cage was mostly intact, the collapsing towers fell straight down. The center structural column could not support the weight of the floors without the outer angle brackets, so it gave way at the affected floors, allowing the upper section of the tower to fall in the only direction it could be expected to fall, straight down. The upper sections brought down most of the cage while the cascading collapse of the floors pulverized the interior and the shear force was enough to bring down the steel supports.
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