The hip hop artist gets a group of live musicians to replay the music they want to use in the style of the original track.
Now they only have to get permission from the publisher of the music, not from the record label or owner of copyright on the original recording. Fees for covering other people’s songs (publisher’s fees) are or were generally much lower than sample release fees (using the actual original recording).
The first rap single to chart – Sugarhill Gang’s “Rapper’s Delight” – used this approach to avoid paying higher fees to Chic’s record label for using Chic’s own original version.