Let's first look at which kind of sources are studied.
Blazars are special type of radio-loud AGN. One of their special properties is that their spectral energy distributions (SED) form the so-called Blazar sequence. They all have a characteristic double peak SED and the position of the peaks mainly depend on their luminosity. The higher the luminosity the lower the peak frequencies (see Fossati et al., 1998 for details)
(image source: Donato et al., 2001)
Leaving details like sub-classes aside, this means that SED of Blazars can be modelled very well, especially if looking at a huge ensemble of Blazars.
Extra-galactic background light (EBL) absorption
Now the high energetic light has to travel from the Blazar to us. The Blazars being far away, much can happen during this travel, notably it can interact with the extra-galactic background light (EBL absorption). If a high energetic photon from the Blazar encounters an EBL photon in the optical or ultraviolet range they can pair produce an electron/positron pair (this is the dominant process for the involved energies).
These charged particles will most likely be subject to magnetic fields or other interactions diverting them from their direct way towards earth.
This means not all of the high energy photons from the Blazar reach Earth and how large the absorption is depends on:
- the distance between the Blazar and Earth
- the energy of the high energy photons from the Blazar
- the energy of the interaction partners from the EBL
Bringing both together
Fermi looked at a large sample of SEDs from Blazars of various luminosities and grouped them based on their place in the Blazar sequence. For each group they could compare the expected SED from the Blazar model with their measured SED and thus determine how large the EBL absorption is depending on the energy. This allows to calculate the density and energy distribution of the EBL.
Knowing the spectral energy distribution and density of the EBL allows to go even one step further and make assumptions on where the photos in the EBL came from. How many stars it took, how long they had to emit photons etc.
As if this alone would not already be interesting enough, Fermi even repeated this for samples of Blazars at different distances, thus probing the EBL at different ages of the universe (remember that we observe the light that reaches the Earth now, thus the light from Blazars further away was emitted earlier than the light from "nearby" Blazars)
(image source: https://fermi.gsfc.nasa.gov/science/eteu/ebl/)