Why is it when you end a relationship with someone like the government or business, they make you sign a form?

Do you read the pieces of paper you're signing? Chances are, if you did, you'd know why you were being asked to sign them. There's usually a reason for the paperwork. Limiting liability, limiting your ability to talk **** after you leave, preventing you from stealing their business/technology, providing you with severance money, etc. It's not so strange that governments and businesses would want clear records. 

So that they can prove it later.

Because if the relationship isn't officially terminated in writing, what's to stop you from going back and not only saying you want their services, but demanding them? They show you what you signed and that's that.

Actually, they make you sign so that peradventure if you decide to come to the business relationship or whatever. I was forced to sign under the 6 months trial when I got my first job. I didn't like the job so I was forced to sign the form. Even though I left my job under the contract.

If you don't want to sign, don't sign.

There is no requirement for you to sign anything you dont want to....

That said, if they are offering severance...I bet its contingent on you signing.