Question: How are these images obtained?
Later in the video the narrator says they took the images using ESO's VLT.
03:40 [Narrator] 14. Making these measurements pushed the power of ESO’s Very Large Telescope to the limits.
(Source: ESO transscript)
Over the whole observation period multiple telescopes and imaging instruments were used. Early observations were accomplished using the NTT. Since around the year 2002 the VLT observed Sagittarius A* with the NACO and GRAVITY instruments too.
This graph puts the rotation period of Sagittarius A* in relation to the time and the observing instruments:
Question: What process is it that is believed to be causing that flashing?
ESO provides older footage over a minutes-timescale showing a flare of Sagittarius A* in May 2003: Flashes of light from disappearing matter
ESO Press Video eso0330 shows the detection of a powerful flare from the centre of the Milky Way galaxy. These and other adaptive optics (AO) images (with resolution 0.040 arcsec in the near-infrared H-band at wavelength 1.65 µm) of the central region of the Milky Way were obtained with the NACO imager on the 8.2-m VLT YEPUN telescope at the ESO Paranal Observatory on May 9, 2003. [...] The position of the 15-year orbiting star S2 (cf. ESO Press Release eso0226) is marked by a cross and the astrometric location of the black hole is indicated by a circle.
The cause of the flickering is probably the same for the video sequence you showed.
A October 2018 ESO publication states:
ESO’s GRAVITY instrument on the Very Large Telescope (VLT) Interferometer has been used by scientists from a consortium of European institutions, including ESO , to observe flares of infrared radiation coming from the accretion disc around Sagittarius A*, [...]. The observed flares provide long-awaited confirmation that the object in the centre of our galaxy is, as has long been assumed, a supermassive black hole. The flares originate from material orbiting very close to the black hole’s event horizon — making these the most detailed observations yet of material orbiting this close to a black hole.
ESO also has another video showing the therory with more detail: skip to about 1:00.
Flares associated with Sgr A* are also described in this answer.
Artist's illustration of heated gas orbiting the black hole:
(Screengrab from 1:25 of that video)
Figure 1 from Do et al. 2019
Figure 1. Top row: a series of K' images taken on 2019 May 13 centered on Sgr A* showing the large variations in brightness throughout the night. The first image from the left is the brightest measurement ever made of Sgr A* in the near-infrared. Also labeled are nearby stars S0-2 (K' = 14 mag) and S0-17 (K' = 16 mag) for comparison. Bottom panel: K' (black) and H-band light curves of Sgr A* from 2019 May 13. On this night, we alternated between H and K' observations. The H-band magnitudes are offset using H − K' = 2.45 mag. There appear to be no significant color changes during the large change in brightness. Red circles show the location of the four images in the panels above.