Speaking as an astronomer: what an oddly pointless (even misleading) press release.
Non-expert readers could be forgiven for reading that and thinking this is some sort of "discovery" ("Hubble observes something new and unusual!"), that the forbidden emission lines only appear in Type 2 Seyfert galaxies ("The spectral lines that Type-2 Seyfert galaxies emit are associated with specific 'forbidden' emission lines."), and that they only come from the active nucleus ("in the midst of an incredibly energetic galactic core"). But none of that is true.
Forbidden emission lines are ubiquitous in astronomy. Any time you see an object with emission lines, some of the emission will be forbidden. So we see forbidden emission lines from Seyfert 2 nuclei, from Seyfert 1 nuclei, from quasars, from non-Seyfert nuclei, from the nebula of star-forming regions, from planetary nebulae, from supernova remnants...
You could pick literally any spiral galaxy that Hubble has observed and title the image "Hubble sights a galaxy with 'forbidden light'", and the title would be just as correct. Or swap "nebula" for "galaxy" and use any Hubble image of a nebula in our own galaxy. (There's even forbidden emission coming from the solar corona.)
Individual atoms or ions can be excited to "meta-stable" states, from which they can decay via emission of forbidden light, by other photons (usually X-ray or UV) or by energetic collisions with other particles.
It has nothing to do with black holes, except that accretion disks around black holes can, due to their high temperature, produce the necessary energetic photons. (But so can massive stars, for example.) The rotational energy of the black hole is not involved.
Not a very well written press release, but an interesting discovery. The Wikipedia article on Seyfert galaxies is well written and provides abundant context: https://en.wikipedia.org/wiki/Seyfert_galaxy
That's a useful article, which includes a more mundane example of "forbidden" light: "Phosphorescent glow-in-the-dark materials, which absorb light and form an excited state whose decay involves a spin flip, and is therefore forbidden by electric dipole transitions. The result is emission of light slowly over minutes or hours."
according to usual approximations (such as the electric dipole approximation for the interaction with light), the process cannot happen, but at a higher level of approximation (e.g. magnetic dipole, or electric quadrupole) the process is allowed but at a low rate.
Critically, they usually involve forbidden intermediates in an approximation of Feyman diagrams where the degree of the graph with quantum interaction nodes/vertices between particles/edges is bounded.
The way out is contracting the edge of the forbidden intermediate between the interactions you'd have to decompose the forbidden transition into, until it's so short in time that you hit Heisenberg or similar and basically quantum-tunnel across the sequence of decomposed interactions.
Which, just like quantum tunneling, can and does happen, but it requires multiple low-occurrence events to randomly happen so quickly in succession that the quantum coherence time is essentially long enough to span across these individual events that have to occur in sequence.
A plot point in Neal Stephenson's Anathem involves the specific wavelength of red light that is seen in a certain circumstance, which is similarly 'forbidden'. (Side note: Had to read Anathem multiple times to really get the novel, but it was worth the investment to me.)
Non-expert readers could be forgiven for reading that and thinking this is some sort of "discovery" ("Hubble observes something new and unusual!"), that the forbidden emission lines only appear in Type 2 Seyfert galaxies ("The spectral lines that Type-2 Seyfert galaxies emit are associated with specific 'forbidden' emission lines."), and that they only come from the active nucleus ("in the midst of an incredibly energetic galactic core"). But none of that is true.
Forbidden emission lines are ubiquitous in astronomy. Any time you see an object with emission lines, some of the emission will be forbidden. So we see forbidden emission lines from Seyfert 2 nuclei, from Seyfert 1 nuclei, from quasars, from non-Seyfert nuclei, from the nebula of star-forming regions, from planetary nebulae, from supernova remnants...
You could pick literally any spiral galaxy that Hubble has observed and title the image "Hubble sights a galaxy with 'forbidden light'", and the title would be just as correct. Or swap "nebula" for "galaxy" and use any Hubble image of a nebula in our own galaxy. (There's even forbidden emission coming from the solar corona.)