The motivation for using Karp-reductions in the theory of $\mathcal{NP}$-completeness
Like Turing reductions, many-one reductions came into complexity theory from computability/recursion theory literature. Cook and Karp reductions are natural complexity theoretic versions of similar existing reductions in computability.

There is a intuitive way of explaining many-one reductions: it is a restriction of Turing reductions where we can ask only a single question from the oracle and the oracle's answer will be our answer.

Now the question is why would we need to study this (and any other kind of reductions like truth-table, weak-truth-table, etc.)?

These reductions give a finer picture than Turing reductions. Turing reductions are too powerful to distinguish between many concepts. A very large portion of computability theory is devoted to the study of c.e./r.e. degrees. The notion of an c.e. set is central. We can have TM machine that can enumerate an infinite set, we may not be able to enumerate its complement. If you want to study c.e. sets then Turing reduction is too strong since c.e. sets are not closed under it. So many-one reductions are a (and maybe the) natural way of defining reductions for this purpose.

Other types of reductions are defined for similar reasons. If you are interested I would suggest checking Piergiorgio Odifreddi's "Classical Recursion Theory". It has a quite comprehensive chapter on different reductions and their relations.

Now for complexity theory the argument is similar. If you accept that $\mathsf{NP}$ is a extremely natural class of problems and you want to study $\mathsf{NP}$, then Cook reductions are too strong. The natural choice is a weaker reduction such that $\mathsf{NP}$ is closed under it and we can prove existence of a complete problem w.r.t. to those reduction for $\mathsf{NP}$. Karp reductions are the natural choice for this purpose.