Ah, where to begin? Sampling and break beats and all that stuff. I'm also a layman. Watching the movie Scratch would clarify it. So would talking with urbanhack.
I think that Paul's Boutique, and the Beasties in general, are considered at the avant-garde of legal sampling. Sampling involves taking a sound recording of another and using it in your own songs. Generally, it's done in snippets, like a break beat (a drum loop) or a bass line.
Here is copyright law, 17 U.S. Code Section 106:
Subject to sections 107 through 122, the owner of copyright under this title has the exclusive rights to do and to authorize any of the following:
(1) to reproduce the copyrighted work in copies or phonorecords;
(2) to prepare derivative works based upon the copyrighted work;
(3) to distribute copies or phonorecords of the copyrighted work to the public by sale or other transfer of ownership, or by rental, lease, or lending;
(4) in the case of literary, musical, dramatic, and choreographic works, pantomimes, and motion pictures and other audiovisual works, to perform the copyrighted work publicly;
(5) in the case of literary, musical, dramatic, and choreographic works, pantomimes, and pictorial, graphic, or sculptural works, including the individual images of a motion picture or other audiovisual work, to display the copyrighted work publicly; and
(6) in the case of sound recordings, to perform the copyrighted work publicly by means of a digital audio transmission.
Why the Beasties? The Beasties sort of brought this method, often used in New York, to the fore in the late eighties and wound up in court a lot because of it. They not only got sued, but they clarified a lot of judicial thinking about copyright law. For example, not many people know that both the actual sound recording and underlying composition of a musical act is protected by copyright law. I forget what case it is, but they got sued for both the sound recording and underlying composition. Their suits can be found here, ranging a decade in time:
https://www.billboard.com/articles/news/65708/beastie-boys-emerge-victorious-in-sampling-suit
and here:
https://www.billboard.com/articles/business/6509648/beastie-boys-lawsuit-pauls-boutique-sampling