There are two main reasons why banks often failed during economic crises in this time period.
First, banks were weaker during economic crises because they often lost money on loans they made. When people deposit money in banks, the banks do not simply hold on to that money for them. Instead, the banks make money by loaning out the money people deposit. The problem comes if the people who borrow the money cannot repay it. If that happens too often, the banks can get in trouble because they have lost the money that people have deposited. This weakens them and could potentially cause them to fail. This can happen more often during economic crises because more businesses that borrow money cannot repay the loans due to poor economic conditions.
The second reason why banks often failed is related to the first. During an economic crisis, people might start worrying about whether they will be able to get their money back from the bank in which they deposited it. If too many people start to worry, they might all go to their banks and demand their money. The banks have to give them their money back because that money belongs to them. If too many depositors come and demand money, they might demand more than the bank actually has (because the bank has loaned it out and the loans have failed). At that point, the bank will have to close because it is out of money. (People no longer do this today because their deposits are insured by the FDIC so they do not have to worry. They will get their money back even if their bank fails.)
Because of these two factors, banks sometimes closed during past economic crises.