To understand "Acceptance for Value" you must first understand contracts. For this purpose we will stick with the basics. There are two types of contracts: Simple and Adhesion.
A simple contract is one in which two parties negotiate an agreement. It is bilateral in that each party gives up something and gets something of equal value in return. It normally is purely up to the parties' discretion as to what the value is. If a dispute develops over the interpretation of the contract, the courts will weigh the evidence, and whichever way the scale tips (however small the tip), then the heavier side wins the game.
An adhesion contract is also bilateral, but it is offered on a "take it or leave it" basis, with no opportunity for negotiating the terms. Insurance and lease contracts are good examples of adhesion contracts. To protect the offeror, every detail of the contract must be specified. If a dispute develops over the interpretation of the contract, the courts will decide all points in the contract in favor of the offeror, and will decide all points not covered in the contract to be in favor of the offeree.
Let's say that you hire someone to paint your house. You have specified that it will be painted yellow, that the job will be finished in three days, and that the painter will notify you when it is done. The specifications are not negotiated because you offered to the painter the contract on a take-it-or-leave-it basis.
The painter accepts the contract and paints the house exactly as specified, within three days, and notifies you that it is finished as specified.
Then...he presents his bill. It is for 3 times as much as you were expecting. You decide to go to court.
The court looks as the contract and notes that the adhesion contract was completed according to specification, but has nothing in it about the value. Because you, the offeror, failed to specify the value, the value decision must be in favor of the painter, no matter what is fair. The painter accepted your contract for the value he perceived, and you pay.
If the contract has been a negotiated contract instead of an adhesion contract, the court would have listened to both sides equally, then make a decision as to what would have been a fair value to pay.
When a traffic ticket is offered by the State through its officer, it is an adhesion contract offered on a take-it-you-can't-leave-it basis. The terms are specified and you have no opportunity to negotiate: You must perform to their specification and satisfaction.
That's their side of the deal. Lucky for you, in their enthusiasm to specify what you must do and give up, they failed to specify what they must give up (i.e. pay) for the privilege of commandeering your life. You accept their offer for the value that it has, and you now specify that value because they failed to specify it in their adhesion contract.
When you accept the contract for value, both sides must fulfill their obligations. In the case of a traffic ticket, you MUST pay the fine, get a driver's license, fix the license plate, or whatever it calls for. On the other hand, the State must pay you the value you placed on the ticket, whatever that value may be. Failure by either side to perform its obligation constitutes an actionable breach of contract.
See What Does Accepted for Value Mean? for a more technical explanation based on UCC and other resources. Also, one person's suggestion for endorsement--USE WITH CAUTION!