This matters because it allows insurance companies to spread risks out over large pools of customers.
The assumption of independence of risks holds that there will not be clustering of a certain kind of risk. In other words, there will not be clusters of houses that are more likely to burn down than others or clusters of people who are more likely to die young. This is important because it allows the spreading of risk. If there is independence of risks, then the insurers can get a large pool of customers who are not likely to all have the same problem occur. Because of this, the companies can get enough premiums to pay off any claims that are made.