My daughters father wants to sign his rights away. What papers do I got to have for him to do so?

Well you will have to file in court, and see if a judge will agree. Most often the judge will not allow them to sign away his rights, and of course he does realise that if he signs his rights away, he still has to pay child support

What does the father think he will get by doing so? Why would you allow it?

Generally, a termination of parental rights is granted only to clear the way for an adoption or if the parent represents a danger to the child.

P.S. I've moved this to the Family Law forum which has hundreds of threads asking a similar question.

As said you will learn an ample amount by reading other threads on this same issue. Also as said it isn't likely that his rights will be terminated. You can't just fill out a paper and be done with it. There is also a pretty good chance that you will have to get a lawyer involved, I have seen judges say they don't have a problem granting a TPR but the paperwork needs to be drawn up by an attorney before he will sign off on it. It depends what state you are in as to if child support will continue or not but no state wants more people living off government money so petitions for a TPR are not taken lightly.