"Why string theory?", you ask. I can think of three main reasons, which will of course appeal to each of us differently. The order does not indicate which I consider most or least important.
A full theory of quantum gravity - that is, a theory that both includes the concepts of general relativity and those of quantum field theory - has proven elusive so far. For some reasons why, see e.g. the questions A list of inconveniences between quantum mechanics and (general) relativity? and the more technical What is a good mathematical description of the Non-renormalizability of gravity?. It should be noted that all this "non-renormalizability" is a perturbative statement and it may well be that quantum gravity is non-perturbatively renormalizable. This hope guides the asymptotic safety programme.
Nevertheless, already perturbative non-renormalizability motivates the search for a theoretical framework in which gravity can be treated in a renormalizable matter, at best perturbatively. String theory provides such a treatment: The infinite divergences of general relativity do not appear in string theory due to a similarity between the high energy and the low energy physics - the UV divergences of quantum field theory just do not appear. See also Does the renormalization group apply to string theory?
Restricting the landscape of possible theories, "naturalness"
Contrary to what seems to be "well-known", string theory in fact restricts its possible models more powerfully than ordinary quantum field theory. The space of all viable quantum field theories is much larger than those that can be obtained as the low-energy QFT description of string theory, where the theories not coming from a string theory model are called the "swampland". See Vafa's The String Landscape and the Swampland [arXiv link].
Furthermore, there are many deep relations between many possible models of string theory, like the dualities which led Witten and others to conjecture a hidden underlying theory called M-theory. It is worth mentioning at this point that string theory itself is only defined in a perturbative manner, and no truly non-perturbative description is known. M-theory is supposed to provide such a description, and in particular show all the known string theory variants as arising from it in different limits. To many, this is a much more elegant description of physics than a quantum field theory, where, within rather loose limits, we seem to be able to just put in any fields we like. Nothing in quantum field theory singles out the structure of the Standard Model, but notably, gauge theories (loosely) like the Standard Model appear to be generated from string theoretic models with a certain "preference". It's hard to not get a gauge theory from string theory, and generating matter content is also possible without special pleading.
Regardless of what the status of string theory as a fundamental theory of physics is, it has proven both a rich source of motivation for mathematicians as well as providing other areas of physics with a toolbox leading to deep and new insights. Most prominent among those is probably the AdS/CFT correspondence, leading to applications of originally string theoretic methods in other fields such as condensed matter. Mirror symmetry plays a similar role for pure mathematics.
Furthermore, string theory's emphasis on geometry - most of the intricacies of the phenomenology involve looking at the exact properties of certain manifolds or more general "shapes" - means it is led to examine objects that have long been of independent interest to mathematicians working on differential or algebraic geometry and related field. This has already led to a large bidirectional flow of ideas, where again Witten is one of the most prominent figures switching rather freely between doing things of "pure" mathematical interest and investigating "physical" questions.