Tl;dr: Bleach is a salt with one of the ions being unstable. When that ion decomposes, the resulting oxygen and chlorine are disruptive to other chemistry.

Salts refer to the type of bond involved -- ionic bonds. Typically a salt is a positive ion and a negative ion that just sort of stick together due to their charges. These bonds aren't very close, and a salt molecule is easily dissolved in water. Once in water, the ions just sort of mix freely with the water molecules.

So here's the thing. Household bleach is sodium hypochlorite, which is technically a salt. They just sell it already dissolved in water at the right strength. The sodium ions in it is identical to the sodium ions in table salt. But the hypochlorite is the key here. This ion is made of a single oxygen and a single chlorine bound to one another. The hypochlorite isn't actually that stable (the solid form could be used as an explosive, actually), and in the presence of other molecules, tends to break down releasing oxygen and chlorine, neither of which are stable by themselves and will prefer to bond to something immediately. Both oxygen and chlorine are strongly electronegative and will bind fast and hard to other organic materials in such a way that they disrupt those materials. After the materials are disrupted, they tend to dissolve easier in water for removal.

Tangent: most household bleach has a significant amount of sodium chloride in it, as a byproduct of the manufacturing process. And it isn't worth it to purify the sodium chloride out of it so they just leave it in there.