Credit: NASA/DOE/Fermi LAT Collaboration
Symmetry breaking has a nice article explaining that the bright flashes are blazars:
Some of the most violent energy sources in the universe, blazars are galaxies that emit jets of particles traveling near the speed of light. In a blazar, one of these jets is oriented directly toward Earth, creating a very strong signal in many wavelengths—including gamma rays.and the dimmer but constant red dots are pulsars:
The crushed cores left behind when massive stars explode, pulsars spin rapidly and sweep a lighthouse-like beam across the sky. When this beam is oriented so that it shines on Earth, we observe it to blink on and off as the star spins.
It’s funny, but we [the Large Area Telescope collaboration] consider pulsars steady sources,” Digel says. “Unlike blazars, they don’t change in brightness, they only pulse.” Because the slowest gamma-ray pulsars flash a few times per second, their on-and-off nature isn’t visible in the highly compressed time of the movie. But in the telescope’s complete data, the flashes are quite clear; in fact, the Large Area Telescope was the first telescope to discern that one of these sources, LAT PSR J1836+5925 (the one on the left edge of the movie), is in fact a pulsar. Previously, it was known as a steady, unidentified gamma-ray object.