"Three" can mean "a set of three things". As such it is a regular noun, with a regular plural. We are speaking in general, so we use a plural.
I don't think there is any particular reason why we say "three". Perhaps when two bad things happen with just think it is coincidence, but when three bad things happen, we think there must be some magical "bad luck" at work. It is just a superstition.
Perhaps a person who believes this superstition sees something bad. Then something else bad happens. Because they have the superstition, they now look for another bad thing. Because they are looking for something bad, they find it! Now their superstition is confirmed. If a fourth bad thing happens - that is interpreted as the start of a new group of three. They remember the world according to their bias. The person who thinks bad things happen in pairs will interpret the same events according to their own bias.
Other superstitions use other numbers without there being any particular reason "Seven years bad luck" "Unlucky 13".