All kinds of music recording require a producer. It is not specific to hip-hop.
Do you know what a movie director is? Movies require a screenwriter, actors, and camera-operators, but they also require a director. The director is the person who manages everything and provides the final word in artistic direction.
A producer is to a music recording what a director is to a film.
It varies a great deal from recording to recording, but a producer is generally paid to manage a recording session or album. They either take a fee up front or they earn a percentage or royalty of the proceeds from the recording. A producer is in charge of renting the studio, paying for session musicians, managing the schedule, and keeping the project on schedule and on budget.
The producer is often the final authority on all the decisions about what songs are recorded, how they are arranged, and how they will sound. If the music you are listening to sounds a certain way, it is because the producer decided that is how it should sound. The producer put together a team of musicians to record that sound that he wanted, and the producer got the project completed.
Producers are particularly important for "solo artists" who don't have a band. A producer might select each of the songs on an album (written by many different writers), come up with musical arrangements (often with the help of a professional arranger or beat-maker) hire all the musicians backing up the artist, and personally supervise every step of the recording process, approving or disapproving of various steps and doing things over again until they get them right. Producers are usually also involved in engineering the recordings and mixing them down.
In many forms of pop music and in many cases the end result and the "sound" depends much more on the producer than on the "artist" or band.
If there is a record label involved that is advancing many thousands of dollars to an artist or act to finance a recording, then the producer is under contract to the record label and is given the task of making sure that the final recording is delivered in a fashion that pleases the record label and the people advancing the money, so they can sell the recordings. Reliable producers who can deliver hit records can become quite wealthy, more wealthy than the "artist" whose name is on the record.
Laurent, I suspect producers are mentioned more prominently in hip-hop because hip-hop producers often make the beats themselves. We used to call them "DJ"s, short for "disk jockey". Thus, the producer is making almost all of the music himself while the rapper is, well, just rapping. And it also has to do with image and marketing. If fans see "Dr. Dre" or "Jay-Z" on the label, they care more about that name than they do about the name of the "artist" rapper. This is just my suspicion; I don't know much about hip-hop.