The bright region is known as a "photon ring". It is light that is heading towards us from a radius of around $1.5 r_s$, where $r_s = 2GM/c^2$ is the Schwarzschild radius of the black hole. So yes, the light is certainly coming to us from the immediate environs of the black hole.

The light travelling towards us is warped by the distortion of space-time caused by the black hole. The warping acts like a magnifying glass meaning we see the photon ring as larger - with a radius of $2.6r_s$.

The reason that we see a ring at all is because the plasma surrounding the black hole is "geometrically thick, but optically thin" at the 1.3 mm wavelengths used in the observations. What this means is that mm-waves are generated by fast-moving electrons in the plasma that is being accreted onto the black hole and the plasma exists over the whole of the region imaged (and beyond), but that most of the emitted light will escape self-absorption.

The latter property is key. When viewing such a plasma, the brightness will depends on the density of the plasma and the path length of the sightline we have into it. This matters greatly near a black hole, because the densest plasma will be nearest the black hole but any light that is emitted and heads inside the location of the "[photon sphere][1]" at $1.5 r_s$ will end up in the black hole, possibly after orbiting many times. Light emitted outwards from dense plasma inside or at the photon sphere may orbit many times and then escape from the edge of the photon sphere. The result is a *concentration* of light rays that appear to emerge from the photon sphere and which we view as a circular ring. The ring is intrinsically narrow but is made fuzzy in the Event Horizon Telescope images by the limited (but amazing) instrumental resolution.

Inside the ring is relative darkness. There is light coming towards us from this direction - from plasma between us and the black hole, but it is much fainter than the concentrated light from the photon ring. Much of the light that would have come to us from that direction has fallen into the black hole and hence it is referred to as the "black hole shadow".

The ring and the shadow should (according to General Relativity) be perfectly circular for a non-spinning, spherically symmetric black hole. The spherical symmetry is broken for a spinning Kerr black hole and small ($\leq 10$%) departures from circularity might be expected (e.g. see [section 9][2] of paper VI in the Event Horizon Telescope series on M87).


  [1]: https://en.wikipedia.org/wiki/Photon_sphere
  [2]: https://iopscience.iop.org/article/10.3847/2041-8213/ab1141#apjlab1141s9