Collisions of a state with other particles, present at finite density, influence the life-time of the state as energy can be transferred to those other particles during inelastic collisions (i.e. they can change state). This change in life-time is related to a change in the width via the uncertainty relation and gives the thermal width.
There is a physical effect known as "collisional broadening" that affects the emission spectrum of gasses at finite density. It occurs because the atoms, during collisions with other atoms of the gas, undergo small temporary changes of the energy levels of their atomic shell. Elastic collisions cause a shift of the spectral lines (thermal mass) and inelastic ones their spectral broadening (thermal width). It can be discussed quantitatively in simple terms and might be the best way to understand these effects. See e.g. Demtroeder - Laser Spectroscopy.
If you are looking at your problem in terms of QFT the analogy is that the states in the medium are not the free quantum states but deformed effective ones that have a different mass and lifetime due to the cumulative interaction with other states in the medium. Colissions with other particles can make a state decay and therfore alter its natural vacuum width. Technically, the real and imaginary parts of a thermal self-energy correspond to the effective mass and in-medium width as you write.