How can antibiotics damage the ear?
This phenomenon is called Ototoxicity, which literally means "toxic for the ear". Mostly the cochlea or the auditory nerve are affected and almost all these cases are connected to medications as gentamicin or cisplatin.

The reasons for this are that the cells are either driven into apoptosis or necrosis. This is caused by destroying mitochondria and releasing calcium and cytochrome c which triggers apoptosis or by deregulating the the JNK-signal transduction pathway and causing necrosis. This image gives a nice overview:



For further reading you can start with the Wikipedia page on ototoxicity and then also read these two papers which are both freely available (the figure above is from the first):


Intracellular mechanisms of aminoglycoside-induced cytotoxicity.
Gentamicin rapidly inhibits mitochondrial metabolism in
high-frequency cochlear outer hair cells.


To answer the question why this occurs in the ear, it seems that there is some genetic predisposition for this, see this paper: "Aminoglycoside-induced ototoxicity." Mitochondrial damages leading to hearing loss is also not uncommin, see here. Noise seems to play an additional role here and is probably the "last drop" to cause hearing loss.