When the World Trade Center towers collapsed they produced many tons of dust, powder, and granular material. This was from two main sources: drywall and concrete. After the initial collapse the bottom half of the core of the building was still standing. It was covered in tons of this material. As parts of it toppled and fell they left trails of this dust.
Some people think that the core was destroyed with tons of carefully placed and detonated pyrotechnics. They say that these trails are evidence of this, as they leave white smoke. There's a lot of reasons why the "explosives" theory would not work, but that aside, the simplest explanation for these trails is that it's dust. We know there was a lot of dust (lower Manhattan was coated in it). That giant cloud you see flowing around World Trade Center (the tall brown building on the right) is all this dust.
However in discussing this with some 9/11 "Truth" activists, they said that dust would not leave a trail like smoke. So I did some experiments to check this. I used cold ash from a fire for the dust. This is probably finer grained than you would see on 9/11, but then the scale is smaller. I dropped a scoop of dust on top of a hammer, held it up in the air, and then let it topple.
As you can see it leaves a trail, and it looks like it's burning. This demonstrates that the simplest explanation for these trails is just dust. No nanothermite required.
A bit more detail: Here's the dust I used:
That was from an earlier attempt, but the brick was not falling fast enough yet when it hit the ground. So only a short trail had formed.
I also tried using the 2 foot square of wood. One of the things people think is that the dust would instantly blow off something falling. But in fact it does not - initially it falls with the object, and then gradually gets stripped away. This is because the object itself shields the dust from the air. It's not exactly dragging the dust down - the dust is actually just falling under gravity, the object is pushing the air out of the way. With the square of wood this effect was even more dramatic, and in a six foot drop very little dust was lost.
The falling pieces of the World Trade Center were much bigger than this, and the amount of dust was much greater. So it make perfect sense that they would leave a long trail as the dust was stripped away at the edges by the air flow.