"Why use a prime like, say, 50mm (or 30mm or 35mm) instead a zoom?"
Many are of the opinion (as apparent from some of the above replies) that using a prime helps the photographer to become a better photographer. Well, that is probably very true. Because it forces (and also allows) the photographers to concentrate on "framing" the composition, consider the angle of (natural) light (which creates different angle of shadow), and consider different points of view for the same scene. This is probably the "purest" way to learn photography.
However, there is another consideration. The "purest" may not be the "most suitable" for each and everyone out there. There is also a high chance that someone at the beginning of their learning process may loose interest when their photographic angle of view is limited to that of just one focal length.
I started with 50mm prime only. However, I got bored with it soon and had to buy another lens just for variety. Ok, my second lens was also a prime (85mm - prime due to the fact that good zooms where not in my budget then) but it gave me some variety and kept my interest alive. So, I think it is best to buy a prime and an optically good mid-range zoom together.
Unfortunately, 50mm on a cropped sensor does not offer a very "attractive" angle of view. A 35mm or 30mm is much better.
Finally, it is sometimes suggested that one does not need a zoom because with a prime one can "zoom with the feet". Well, that is not entirely correct. One can definitely get the same area (in two dimensional term) of the scene by zooming with the feet (with a wider lens) as one would with a tele lens; but, zooming with the foot changes the "point of view" of the photographer with respect to the scene and hence changes the perspective. This definitely changes the composition of the scene. Therefore, zooming with the feet and zooming a zoom lens is not the same thing - for composition.