An even more excellent question than the opposite one!
The first happy group of people for whom the discovery of SUSY at the LHC would be spectacular would be the experimenters. They would experience fireworks of activity, facing the task to find as many superpartners as possible and to measure their properties.
All their masses - and several other parameters - would be news for the theorists. We would have to learn lots of new, currently unknown numbers from the experiments, the so-called soft SUSY-breaking parameters. It's conceivable that the masses would display some patterns - e.g. mass ratios - that agree with one of the scenarios of supersymmetry breaking. So some of the mechanisms of supersymmetry breaking would be supported; others would ultimately be abandoned.
Figuring out the logic of SUSY breaking
Quite a lot of data is needed from the first hints of SUSY to the point in the previous paragraph because one needs to see many new particles and some of them are inevitably harder than others. However, if this job is completed, we will know whether SUSY breaking is mediated by gravity mediation, anomaly mediation, gauge mediation, mirage mediation, or something else.
It would be a lot of interesting work for the theorists and it is more likely than not that they have already done the essential work for any major type of SUSY breaking that could be observed by the LHC. Obviously, one of the schemes would be studied in much more detail if we knew that it is the correct one. Some knowledge about the mechanism of SUSY breaking - which may be guessed from the superpartner masses - would tell us a great deal about the required underlying compactification of string theory etc.
Also, the role of the newly found SUSY for all the problems that SUSY is capable of solving - that have been discussed under the question
What if the LHC doesn't see SUSY?
(I mean dark matter, gauge coupling unification, and the hierarchy problem) would be studied in much more detail, too. I don't have to enumerate them again because this text would become highly redundant. It's obvious that if the existence of low-energy SUSY were supported by the experiments, all other things that were expected to be solved by SUSY would be studied much more seriously, much more materially, and in much more detail. Some of the advantages of SUSY are only understood superficially - this situation would have to improve.
LHC SUSY discovery and string theory
The discovery of SUSY would be amazing - the first discovery of a new spacetime structure and symmetry since Albert Einstein's relativistic adventures. It would be great, it may happen, and string theorists must be ready to take credit for it. No doubt, at least in the Western (non-Soviet) context, SUSY is a daughter of string theory. First, it was discovered by Pierre Ramond on the 2-dimensional world sheet of a string, before it was exported to the 4D spacetime and other higher-dimensional spacetimes and before it was found in the 10D spacetime of superstring theory as well. Historically, supersymmetry was one of the first amazing ideas that the physicists were forced to discover because string theory led them to discover them.
All the critics of string theory would be proved to be spectacularly wrong - in fact, narrow-minded folks who wanted to prevent the mankind from discovering one of the most fundamental properties of Nature; everyone could suddenly see that they are on par with the geocentrists and they would hopefully never show up again in the public. I would win a USD 10,000 bet against a phenomenologist who agreed with 100-to-1 odds - this "uneven" number itself is enough to show that some of the enemies of supersymmetry resemble a fundamentalist religious sect.
The pragmatic and largely non-anthropic phenomenological attitude to string theory would prevail. People would probably agree that a feature of the vacua doesn't have to be "generic" for it to become true. The anthropic principle would fade away. Because low-energy SUSY would become a fact, people would kind of accept that with some extra knowledge about the reality used as assumptions, supersymmetry is a consequence of string theory. See
Why string theory implies supersymmetry
Lots of knowledge about the likely compactification of string theory would become much more accessible. String theorists - which would become a quickly growing group - would very likely converge to some opinion whether heterotic string theory; heterotic M-theory; M-theory on a G2 holonomy manifolds; type IIA intersecting braneworlds; or F-theory on Calabi-Yau four-folds is the most viable approach to phenomenology.
I have discussed in what sense string theory probably implies supersymmetry. Now, you're obviously interested in the opposite hypothetical implication - whether supersymmetry implies string theory. We can't prove this implication as a mathematical theorem but it would become extremely persuasive. First of all, supergravity (SUGRA) would become an inevitable component of all effective field theories because it follows from general relativity (established) and supersymmetry (hypothetically established in our thought experiment).
In this text,
Two roads from N=8 SUGRA to string theory,
I argue that supergravity suffers from two kinds of problems: non-perturbative inconsistencies; and the unacceptable phenomenological limitations of its maximally supersymmetric N=8 version (which is the only perturbatively finite one). Attempts to fix either of those limitations of supergravity inevitably leads to string theory with its more powerful toolkit. If you want to watch a 30-minute lecture by a Dirac Medalist explaining why supergravity can't be decoupled from string theory and why one needs all of string theory to preserve the consistency, see this November 2010 talk by Michael Greene in Trieste:
Similarly, in the text a few paragraphs above, I argue that locality of general relativity implies that there must exist magnetically charged objects - at least black holes - and the Dirac quantization rule implies that the charges must belong to a lattice. The choice of the lattice is equivalent to the point of the moduli space of inequivalent stringy vacua; the noncompact symmetry group of SUGRA is inevitably broken down to its discrete exception subgroup, the U-duality group. In the appropriate limits of the moduli space of the inequivalent vacua, we may derive the existence of objects known from string/M-theory as well as their excitations, and we may pretty much complete the rest of string theory by consistency arguments.
Green-Schwarz anomaly cancellation
There is one more characteristically stringy structure that would become necessary if we add one more assumption: the Green-Schwarz mechanism that mixes tree-level terms with one-loop terms in a very stringy way - originally discovered by Green and Schwarz in 1984 when their discovery sparked the first superstring revolution.
This mixing of contributions at different orders is extremely unnatural in perturbative field theory. And we would have evidence that it takes place in Nature assuming that there exists at least one axion - or, in SUGRA terms, at least one linear supermultiplet. If there are axions, which may be needed to solve the strong CP-problem, there are also new kinds of anomalies (in particular, the "conformal anomaly" of supergravity) analogous to the 10-dimensional anomalies addressed by Green and Schwarz in 1984. A 4-dimensional version of the stringy Green-Schwarz mechanism would be needed to cancel those anomalies.
Theorems may be rigorously proved in the highly symmetric vacua only but not in the real world. However, in the real world, the evidence that string theory is right would become overwhelming.