Well, I suppose there are two ways to skin this particular cat: the hard way and the very hard way. As you’d expect, supersymmetry (SUSY) did emerge from versions of “our current understanding of the strong/weak/em/gravity forces.” However, as Steven Weinberg has said, with a slight degree of overstatement, the process by which they did “is as peculiar as anything in the history of science.” That’s the very hard way.
Even more than usual, most technical presentations of the concepts thus avoid the historical route, starting instead from something called the Coleman-Mandula theorem. This is the merely hard way. In practice, it’s very formalistic - it’s about looking for patterns in the maths and thinking up elegant variants on them. Theoretical physicists are comfortable with this, but it does make it difficult to explain. Particularly without equations.
So what’s the Coleman-Mandula theorem ? It’s all about possible symmetry groups. In particular, it’s about two different types of them. One is the Poincare group. But all that’s jargon. In plainer English, the theorem assumes that special relativity is true. The more interesting issue is the other type of symmetry group: the so-called internal one. It’s probably easiest to take an example, the strong interaction. The modern, post-Seventies, view of the strong interaction is that it’s due to quarks exchanging gluons. There are three types of quark, identified by different colours: red, green and blue. When a quark exchanges a gluon with another quark both change colour, because the gluons themselves carry colour charge. Now when you describe this in a quantum field theory (QCD), the mathematics of the theory are those of a symmetry group. And like all groups, the central, rather simple, idea is that of operations. In this case, the operations are almost trivial: for instance, the act of changing a blue quark into a green one. Suppose the quark emits a blue-antigreen gluon. It loses it’s blueness and has to aquire greenness to compensate for the antigreenness of the gluon. So the blue quark becomes green. That’s the operation of changing a quark of the blue type into one of the green type. Since this operation is done by emitting a gluon that will be absorbed by another quark, the symmetry group is very tied into the interaction between quarks - they’re really the same thing. [To avoid confusion, I should note that you don’t get blue-green gluons. There has to be a bit of anti-ness involved.]
Basically, what the Coleman-Mandula theorem says is that if special relativity is true, you’re very restricted in your choice of internal group, i.e. the allowed form of the interaction. In particular, the operations of the group can only interrelate particles of the same spin. That’s not a problem in QCD, since all quarks are spin 1/2. But the theorem would disallow a version of QCD in which one of the three types of quarks had spin 1. Such a type of quark wouldn’t be allowed to interact with the spin 1/2 type quarks in the way they did between themselves.
Except that Coleman and Mandula were wrong (for once). They’d made the, seemingly safe, assumption that the operations of the symmetry group can only relate bosons to bosons and fermions to fermions. (Bosons are particles with integer spin and fermions are ones with non-interger spins. Which leads to other stuff.) But allowing operations that turn fermions into bosons, and vice versa, is a loophole. So what happens if you allow such operations ? Supersymmetry ! You can prove that a sort of super-Coleman-Mandula theorem. If special relativity is true, then the only possible symmetry groups relating particles of different spins are SUSY ones. And they also relate fermions to bosons. So you could have a theory in which one of the (s)quarks had spin 1, but not one in which it had spin 3/2. The first requires an operation relating a spin 1/2 fermion quark to a spin 1 boson squark. Allowed. The second relates the spin 1/2 fermion quark to a spin 3/2 whatever that’s also a fermion. Not allowed.
Now, in general, theoretical physicists are the sort to believe that if something is possible, it’s probable. So the fact that SUSY theories aren’t disallowed by the super-Coleman-Mandula theorem means that people started to look at their consequencies. It’s basically the feeling that Nature wouldn’t use something like QCD and then turn her nose up at SUSY, when both are just as possible. Three decades of work have thrown up plenty of cases where SUSY would be theoretically nice. It’s also thrown up preciously little experimental need for the idea. Even today, there’s really only the one case - to do with the unification of “running couplings” at very high energies - where SUSY is easily the better explanation. Some of us have even been known to niggle the True Believers that it will never be found …
And the very hard way ? Messy, messy, messy. For a start, the idea arose independently in the USSR and the West. And in the West, it came, oddly enough, out of String Theory. The usual way of telling the story is that it was saved by SUSY, whereas in fact it invented it, it was imported to the mainstream, then was used by string theorists, whereupon String Theory became the fashion (in the mid-Eighties).
The Coleman-Mandula theorem was a reaction to particular attempts to explain the strong interaction in the Sixties. Those attempts were superceded by QCD anyway and it was only after SUSY had been invented that people noticed the connection.