The more that we human beings are protected from the consequences of our actions, the more risks we are likely to take. Think about it this way: if you were going to try to do a trapeze act, wouldn’t you do a much fancier act if you had a safety net under you? The same goes for federal deposit insurance.
When people deposit their money in banks, the banks do not just let it sit there. Instead, they make investments. As is typically the case, riskier investments also have the bigger potential rewards. This means that banks can potentially make more money by taking big risks than they could by playing it safe. With federal deposit insurance, they are protected to some degree from the consequences of those risks. If they take too many bad risks and the bank fails, all of the depositors will get their money back. This considerably reduces the consequences of taking bad risks.
It is for this reason that we can say that federal deposit insurance creates a bit of a moral hazard by making it more likely that banks will take big risks.