I'd like to give an answer which is less technical than some of the answers given so far, and more general. One of the main questions in group theory, informally, is just to describe the possible finite groups. "Describe" could be mean classify, and of course there is by now the fantastic Classification of Finite Simple Groups. But let's say that that wasn't yet known --- after all its proof is incredibly complicated. In any case not every finite group is simple. What could a group with 15 elements look like, say? Or a group with 100 elements? If you try to answer questions like this cold, most of them are very hard. Of course, you have various creative methods to give examples of finite groups that do exist: symmetric groups, dihedral groups, matrix groups, etc. However, it is surprisingly difficult to get started with the question of what groups do NOT exist. You would like some structure theorems about a mysterious finite group that someone else may or may not find, and not theorems about explicit finite groups that you make.

One of the few ways to get started is with the Sylow theorems, together with result that every p-group is nilpotent. (The second result follows from the lemma that every p-group has a center.) Sylow's theorems say that if you have a group with 15 elements, then it has a subgroup with 5 elements, and that subgroup is normal. It also has a subgroup with 3 elements, and with a tad more work that I will skip, there is only one group with 15 elements, the cyclic group. Now, finding the groups with 15 elements is not all that exciting, but how else were you going to analyze the groups with 15 elements or any other number of elements? In the specific case of 15 elements, there is Cauchy's theorem that there exists an element of every prime divisor order; but Sylow's theorems are best understood as a much stronger version of Cauchy's theorem.

A semi-separate way to get started is by observing that every finite group has a composition series with simple composants. This reduces the problem of classifying finite groups to classifying finite simple groups, and the problem of composing them together. But then what? To begin to classify finite simple groups, you are back to Sylow's theorems. Soon enough you need other tools, but as Keith Conrad explains, most of these tools use Sylow's theorems directly, while the others use them indirectly.

The above discusses why Sylow's theorems are important in group theory. A different question is why the theorems are important in the rest of mathematics. Occasionally people in other areas have the same classification concerns as group theorists. For instance, one question in 3-manifold topology is to classify which finite groups are fundamental groups of closed 3-manifolds. This question is now settled thanks to Geometrization and Perelman. However, the related question of which finite groups can act freely on a homology 3-sphere is still open, and it uses the same general classification theory.

But most of the time, mathematicians outside of group theory care much more about specific finite groups that exist than finite groups that do not exist. Since the Sylow theorems are at heart non-existence results, it is awkward to sell them to an audience that cares much more about the converse. Like if you want to buy a car, it would be strange for the salesman to brag about a technical paper that certain cars will never be built. Nonetheless, a wide variety of finite groups have turned out to be useful elsewhere in mathematics: In coding theory and combinatorial designs, in Galois theory, in string theory in the case of the monster group, etc. Many of the known finite groups were first considered simply as examples of finite groups that exist. Almost every finite simple group, in particular, has been used outside of group theory. Finite group theorists would not be very competent at finding finite groups for others to use, if they did not also have non-existence results.