How do you see the beam of a flashlight in the dark? Because it hits stuff in the air and bounces off.

And the particles previously in the beam that already hit something are likely to hang around and be candidates for being hit themselves. And eventually they wind up as part of a big glowing cloud that you can see in the pictures. See https://d2r55xnwy6nx47.cloudfront.net/uploads/2021/05/cyga_v... linked in the picture for what it looks like. And for evidence that we can still see the beam when it isn't pointed directly at us.

The radiation is not pointed in the same direction as jets like a laser. There is bulk motion of the plasma in a given direction, but the radiation is emitted from particles that are moving in a different directions within that plasma. So the radiation spread is wider than the jet direction. Interestingly, depending on if the jet we are viewing is pointed towards us or not can impact on the radiation signature in the form of blue shift and red shift. If the orientation is just right, bits of jet appear to move faster than to the speed of light, but it's just a geometric effect.

I’m speculating, but maybe it lights up a conveniently located dust cloud

I believe that this is generally correct. Relativistic particles impinging upon even dilute gas will have bright signatures in many bands.

> Many jets are thin and bright all along their length. “How does it shine? How do we see it?” Chen wonders.

It looks like nobody knows for sure.

Perhaps reflection from other bodies? Or simply some kind of radiation emitted from the jets?