The single best way to conceptually understand what perturbative string theory really is about and why it is a natural concept to consider, is to first understand that ordinary
perturbative quantum field theory has an equivalent formulation that is called the
worldline formalism. In this formulation each Feynman diagram appearing in the computation of the
S-matrix is identified with the correlator of a 1-dimensional quantum field theory, namely with the worldline theory of the first quantized particles that are the given quanta.
This worldline formalism in itself is highly interesting, as it is this formulation that most directly connects quantum field theory to
zeta function regularization and hence to the mathematics of zeta functions and hence to structures famous in number theory. But for the purposes of the present question, of course the following aspect is relevant:
If the perturbation series of a quantum field theory is the sum over all appropriate 1-dimensional graphs of the correlators of a 1-dimensional worldline field theory, then...
...is there a generalization of this where one instead does a sum over d-dimensional spaces of the correlators of a d-dimensional worldsheet theory, as this is an immediate generalization?
And the answer is that doing this for d > 2 seems to be impossible. Doing it for d = 2 gives perturbative string theory.
More exposition that presents string theory along these lines is in