In an isolated molecule /atom the intrinsic width of the transition is limited by the lifetime of the upper level and this can be huge ( seconds) and so the line-width can be minuscule by time-energy 'uncertainty'. If the decay is exponential the spectral line shape is Lorentzian by Fourier Transform theory.
In a medium, the surrounding solvent or gas molecules repeatedly interact with our victim molecule by collisions and by electric and magnetic dipoles etc, and do so in a random fashion and thus cause fluctuation in its energy levels. These changes are then observed in the transition energy and so broaden the observed transition in the ensemble of molecules. So far this is quite general. This paper examines one model of this, linear + quadratic coupling and the quadratic part is presumably why the peak is split - but I only skimmed the paper so you will have to check this. The detailed maths is just the technical part, ignore this and try to get to the basic physics first.
(The term 'thermal bath' is just jargon for all the random interactions from solvent/buffer gas surrounding the victim molecule, 'modulation' just means changes, so the first sentence could read 'We describe how random interactions with a solvent or buffer gas changes the observed spectroscopic line shape ....')