in the late 1960s, the strongly interacting particles were a jungle. Protons, neutrons, pions, kaons, lambda hyperons, other hyperons, additional resonances, and so on. It seemed like dozens of elementary particles that strongly interacted. There was no order. People thought that quantum field theory had to die.
However, they noticed regularities such as Regge trajectories. The minimal mass of a particle of spin $J$ went like
$$ M^2 = aJ + b $$
i.e. the squared mass is a linear function of the spin. This relationship was confirmed phenomenologically for a couple of the particles. In the $M^2$-$J$ plane, you had these straight lines, the Regge trajectories.
Building on this and related insights, Veneziano "guessed" a nice formula for the scattering amplitudes of the $\pi+\pi \to \pi+\rho$ process, or something like that. It had four mesons and one of them was different. His first amplitude was the Euler beta function
$$ M = \frac{\Gamma(u)\Gamma(v)}{\Gamma(u+v)}$$
where $\Gamma$ is the generalized factorial and $u,v$ are linear functions of the Mandelstam variables $s,t$ with fixed coefficients again. This amplitude agrees with the Regge trajectories because $\Gamma(x)$ has poles for all non-positive integers. These poles in the amplitude correspond to the exchange of particles in the $s,t$ channels. One may show that if we expand the amplitude to the residues, the exchanged particles' maximum spin is indeed a linear function of the squared mass, just like in the Regge trajectory.
So why are there infinitely many particles that may be exchanged? Susskind, Nielsen, Yoneya, and maybe others realized that there has to be "one particle" of a sort that may have any internal excitations - like the Hydrogen atom. Except that the simple spacing of the levels looked much easier than the Hydrogen atom - it was like harmonic oscillators. Infinitely many of them were still needed. They ultimately realized that if we postulate that the mesons are (open) strings, you reproduce the whole Veneziano formula because of an integral that may be used to define it.
One of the immediate properties that the "string concept" demystified was the "duality" in the language of the 1960s - currently called the "world sheet duality". The amplitude $M$ above us $u,v$-symmetric. But it can be expanded in terms of poles for various values of $u$; or various values of $v$. So it may be calculated as a sum of exchanges purely in the $s$-channel; or purely in the $t$-channel. You don't need to sum up diagrams with the $s$-channel or with the $t$-channel: one of them is enough!
This simple principle, one that Veneziano actually correctly guessed to be a guiding principle for his search of the meson amplitude, is easily explained by string theory. The diagram in which 2 open strings merge into 1 open string and then split may be interpreted as a thickened $s$-channel graph; or a thick $t$-channel graph. There's no qualitative difference between them, so they correspond to a single stringy integral for the amplitude. This is more general - one stringy diagram usually reduces to the sum of many field-theoretical Feynman diagrams in various limits. String theory automatically resums them.
Around 1970, many things worked for the strong interactions in the stringy language. Others didn't. String theory turned out to be too good - in particular, it was "too soft" at high energies (the amplitudes decrease exponentially with energies). QCD and quarks emerged. Around mid 1970s, 't Hooft wrote his famous paper on large $N$ gauge theory - in which some strings emerge, too. Only in 1997, these hints were made explicit by Maldacena who showed that string theory was the right description of a gauge theory (or many of them) at the QCD scale, after all: the relevant target space must however be higher-dimensional and be an anti de Sitter space. In AdS/CFT, much of the original strategies - e.g. the assumption that mesons are open strings of a sort - get revived and become quantitatively accurate. It just works.
Of course, meanwhile, around mid 1970s, it was also realized that string theory was primarily a quantum theory of gravity because the spin 2 massless modes inevitably exist and inevitably interact via general relativity at long distances. In the early and mid 1980s, it was realized that string theory included the right excitations and interactions to describe all particle species and all forces we know in Nature and nothing could have been undone about this insight later.
Today, we know that the original motivation of string theory wasn't really wrong: it was just trying to use non-minimal compactifications of string theory. Simpler vacua of string theory explain gravity in a quantum language long before they explain the strong interactions.
