how is it used in the judicial system ?
They have been applied on issues ranging from the power of the federal government in foreign affairs, to the validity of ex post facto laws.
The Federalist Papers
Federal judges, when interpreting the Constitution, frequently use "The Federalist Papers" as a contemporary account of the intentions of the framers and ratifiers. They have been applied on issues ranging from the power of the federal government in foreign affairs (in "Hines v. Davidowitz") to the validity of ex post facto laws (in the 1798 decision "Calder v. Bull", apparently the first decision to mention "The Federalist"). , "The Federalist" had been quoted 291 times in Supreme Court decisions.

who were the authors of the federalist and what was the purpose of writing it It was written by Alexander Hamilton, James Madison, and John Jay under the collective pseudonym "Publius" to promote the ratification of the United States Constitution. what is it about ? The authors of "The Federalist" intended to influence the voters to ratify the Constitution. In Federalist No. 1, they explicitly set that debate in broad political terms:It has been frequently remarked, that it seems to have been reserved to the people of this country, by their conduct and example, to decide the important question, whether societies of men are really capable or not, of establishing good government from reflection and choice, or whether they are forever destined to depend, for their political constitutions, on accident and force.
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The Federalist Papers [SEP] Judicial use
Maryland", that "the opinions expressed by the authors of that work have been justly supposed to be entitled to great respect in expounding the Constitution. No tribute can be paid to them which exceeds their merit; but in applying their opinions to the cases which may arise in the progress of our government, a right to judge of their correctness must be retained." In a letter to Thomas Ritchie in 1821, James Madison stated that "the legitimate meaning of the Instrument must be derived from the text itself; or if a key is to be sought elsewhere, it must be not in the opinions or intentions of the Body which planned & proposed the Constitution, but in the sense attached to it by the people in their respective State Conventions where it recd.