For a contract to be valid, there needs to be "consideration" given to both sides, which is legalese for "something of value", including money, credit, service or goods. In the case of your neighbors, you are not giving them any consideration
With your bank, you are engaged in a contract, with each of you receiving consideration. Generally, you get consideration in the form of a) safekeeping of your money, b) transfer service of your money (electronically and in the form of checks), and c) interest, however good or pitiful it may be. You bank generally gets consideration in the form of a) any fees you pay and b) the right to use your money in fractional reserve banking to make loans to other people.
Generally, with a bank contract, there is generally no temporal binding on your relationship; by which I mean, there is no penalty (with perhaps the loss of a promotional interest rate) if you were to withdraw all of your funds today without any notice and close your account. (Though some savings account have withdrawal delays that the bank may or may not choose to invoke, but that is more for avoiding "runs" more than anything else).
However, a bank can also choose to stop doing business with you, close your account, and give you your money, ending the relationship. Because you don't have a temporal binding in your contract, it can do so at any time. It can also refuse to do business with you under the existing terms, and only continue business on new terms.
For example, lets say that you have a lawn, and we have a contract (noting that a contract need not necessarily be written down) that I come by every Saturday and Tuesday morning and mow it, for which you pay me $20 a week. At any time, I could refuse to continue under the current terms, and demand new ones, which you could acquiesce to, refuse, or make a counter offer. So if I came to you and demanded $40 dollars a week, you could agree, refuse (and either end the contract or continue offering $20 a week for two days mowing), or counter offer, perhaps offering $30 or my only coming on Saturdays to mow the lawn for the original $20.
EDIT: Note that this process doesn't work with a loan, because with a loan, there is technically only a single transaction: the lender trades the one-time consideration of the loan (money) for the borrower's binding promise to repay the money, by a specified schedule with a specified rate of interest.
You actually have three options here: accept the new terms, send a letter to opt out, or stop using the service and terminate the contract that you have in place with your bank (which should be relatively simple and painless, EDIT: Unless you have a loan, but if you did, said loan would almost certainly not be affected by the change in terms, as one side cannot unilaterally change a loan contract). The reason for the opt out clauses is that binding arbitration clauses are under fire, with many (myself included) believing that they infringe on important rights. The opt-out clause, which gives you the option not be bound by the clause if you take action, is intended to be more palatable to judges, who might otherwise be more inclined to strike the contract in its entirety or otherwise create precedent hostile to arbitration.
As for the required format of the opt-out notice: yes, they are intentionally making it difficult for you to file it; however, they have chosen a format that is judicially recognized and until very recently, was the default means of judicially recognized communication. It was only in the last few years that courts have allowed for text messages, emails, etc. to be used for official court communications, such as summons, subpoenas and other serviced documents. Before that, the only option was mail, preferably certified, or in some cases, newspaper announcements, if mail service could not be used.