For those of us who don’t understand much legalese, what does this mean?
Anyone?
An Amicus Curiae, or "friend of the court," brief is a legal argument filed by someone who is not a party to the suit. They allow parties who will be affected by the outcome of the case, though they are not directly involved, to weigh in.
For example, if there's a suit with first amendment implications against the Washington Post, other publications not named in the case will file amicus briefs. Suits against one state frequently receive amicus briefs from other states. If it's an abortion case, pro-choice and pro-life groups will both weigh in.
The document posted is Bill Anderson asking for leave -- permission -- to present an amicus brief despite the fact that the court's deadline for such filings has passed.