Through oxidative damage, primarily.

Oxygen is very reactive; it tends to suck electrons away from other molecules, a process called "oxidation". When life scientists speak about "oxidative damage", they are generally referring to oxygen atoms pulling electrons away from other molecules, thus destabilising those molecules, to the point that they may change their shape (and thus their function) or even break apart.

Crucially, hydrogen peroxide can't be excluded from cells; it can pass through cell membranes easily. Once inside a cell, it usually breaks down into a hydroxyl radical and a hydroxide ion. The hydroxyl radical (HO+) is extraordinarily reactive, and will steal an electron from almost any organic substrate it comes in contact with. However, this usually turns the oxidised substrate into a radical itself, which will steal an electron from another nearby molecule, initiating a chain reaction of electron-stealing which can destabilise an entire area of a cell. Also, DNA is very susceptible to oxidative damage, and since bacteria have a single chromosome controlling all their life functions, that kind of damage can be disastrous to them. The prokaryotic bacteria often lack repair mechanisms which can limit this kind of damage, too, which is why it often ends with the bacterial cell dying. However, any kind of cell can and will suffer oxidative damage in the presence of free radicals.