How are they formed? And why are they so bright?
Black holes are small relative to their mass, matter that falls into a black hole will accelerate to very high speeds and then get crammed into a very tight volume. If even a relatively modest amount of material is falling onto a black hole it will be bottlenecked like sand falling through the middle of an hourglass. This process creates an accretion disc of matter that is queued up trying to fall into the black hole. Because of the extreme forces involved the friction in the accretion disc is so strong that it heats matter to extreme temperatures. The accretion process is one of the most efficient in converting rest-mass to energy, even fusion only converts a fraction of a percent of rest-mass to energy, but accretion can convert up to 40%.
These enormous temperatures not only create lots of high energy EM radiation but also ionize atoms and create charged particle/anti-particle pairs. Electrically charged particles in motion will create magnetic fields, in a black hole's accretion disc there are a lot of electrically charged particles moving very, very fast, creating extremely strong magnetic fields, which in turn accelerate ions in jets perpendicular to the disc. The formation of the jets themselves is an enormously complex process (since the ions both create the magnetic field and are accelerated by it) that is not well understood. Nevertheless, the enormous energies involved are very favorable to the production of charged particles and accelerating them to high energies in the bipolar jets. They are so bright because the particles within them are at very high energy (particle/particle collisions will release a lot of energy, recombination of ions with electrons will release a lot of energy) and also because of the presence of large amounts of anti-matter (resulting in annihilation reactions which release gamma rays).
$\begingroup$ "Electrically charged particles in motion will create magnetic fields" is true only if there net charge in motion. Plasmas are inherently charge neutral upon creation from charge neutral matter. $\endgroup$ Apr 26 at 13:50