radiation - Why do black holes choke?

I read that a black hole can sometime "choke" on a star: "...the disrupted stellar matter was generating so much radiation that it pushed back on the infall. The black hole was choking on the rapidly infalling matter."

I read the report and thought it was reasonable, and in line with Rob's answer. But note that there's no certainty that this is what actually happened. They saw a very bright "optical transient" event, circa ten times brighter than a normal supernova. See the paper on the arXiv: A Luminous, Fast Rising UV-Transient Discovered by ROTSE: a Tidal Disruption Event? There's a question mark on the end of the title, they don't know for sure. And see this from the news article:

"To narrow it down from four possibilities, they studied Dougie with the orbiting Swift telescope and the giant Hobby-Eberly Telescope at McDonald, and they made computer models. These models showed how Dougie's light would behave if created by different physical processes. The astronomers then compared the different theoretical Dougies to their telescope observations of the real thing."

They came up with what they thought was the best fit. But we don't know for sure that this was actually a black hole "choking" on a star.

Why does a black hole choke?

We don't know for sure. Note in the paper on page 8 and 9 they talk about an off-axis GRB interpretation. It's possible that this is what happened. Or something else altogether. The tidal disruption idea looks like a good fit, but they don't know for certain. Anyway, have a look at the summary on page 12 for a nice round-up:

"The tidal disruption scenario was explored by fitting the event to an amended version of the model presented in Guillochon et al. (2014). The TDE model yielded a good fit to the photometric and spectral evolution of the flare, with the highest-likelihood models suggesting a disruption of a solar-mass star by a black hole."

How do astronomers manage to observe this rare event?

They didn't actually observe a black hole choking on a star. What they saw was an optical transient. Something that looked like a supernova, but didn't fit the supernova pattern. So they think it was the disruption of a star by a black hole.