Banks worry that the large gift might be a loan that is ultimately expected to be repaid. If so, that affects the cash flow of the recipient, and makes it more difficult to make the mortgage payments to the bank.
In some cases, of course, it is an informal loan: Dad advances a large $X to son to use as a downpayent, but does not charge interest and the expectation is that the money will be returned in smaller chunks as and when the son can afford to repay Dad.
In some cases, Dad truly means it as a gift, but son feels an obligation to repay the money, if not explicitly, then by paying for the first few months of Dad's nursing home stay, etc. So, banks like to have an explicit document such as a copy of a letter from Dad saying that this money is a gift, and some assurance that this is on the up and up.
If the amount is larger than the maximum gift that can be given each year without having to file a gift tax return, then some assurance that a gift tax return will be filed is helpful. Mentioning this in the letter is good: it indicates that there are no secret handshakes or secret agreements to the effect that this is in fact a loan, with or without regular repayments.