Two ways: (i) already damaged hair cells, this is inferred from previous sensorineural hearing loss (hyperacusis caused by noise); (ii) through the hidden hearing loss mechanism + brain turning up volume.
In hyperacusis caused by noise the brain looks for a lost frequency. The brain wants to perceive sounds like it did before noise damage. Then it turns up volume for all frequencies trying to get more quality on a frequency that is damaged or irreversibly lost. It's possible to get more volume but not more sound quality, due to noise damage: that frequency is already damaged.
My personal sensation before and after hyperacusis:
Before: when a loud sound happens, the perceived "impact" is lower than after hyperacusis. It is like the sound vibration is distributed among different frequencies.
After: the sound is perceived very loud, but "off key". A high pitched sound can be perceived louder but lower in pitch, or distorted in pitch. My personal feeling is this has to do with cochlear damage.
I sometimes imagine sound is a hammer and hair cells are a bed made of nails. If you have perfect hearing, the bed has a lot of nails, it is very dense, the hammer hits them and the impact is distributed among the nails. If there are less hair cells or damaged hair cells, then you have a "bed of nails made of less nails or made of nails that are bent", and the impact feels stronger, and can cause more damage...
For people with recruitment, the sound perception is also altered. There is hearing loss with recruitment. This means: (i) some very low sounds cannot be heard due to the hearing loss; (ii) some medium volume sounds are perceived ok but not at the right pitch; (iii) louder sounds and loud sounds are perceived louder than a normal person. That's why partially deaf people can tell they do not hear a person and when that person raises the volume they say "don't yell at me, I am not deaf."