The simplest solution does not always imply the correct solution, nor does it imply any statement regarding proof, at least from my understanding.
But what Ockham's Razor does give you is the default or null hypothesis. If you have a set of alternatives to cover a particular set of observations, the hypothesis which has the fewest number of components covering that set of data is, methodologically speaking, the one which has preferred status. The reason is simple:
the simplest hypothesis is going to be the one which is most vulnerable to being falsified. Having fewer `moving parts', it has fewer resources available to evade counterexamples. The more complex the theory is, the more apparatus it can recruit in the face of `apparent' counterexamples.
So take two hypotheses X and Y, where Y properly includes the resources of X . Suppose that the X happens to correspond to the reality we assume holds external to our own desires, beliefs and preferences. Since Y contains X, Y will not be falsified by any observations we can make—and yet, by hypothesis, it is incorrect insofar as it assumes that the extra material it contains over and beyond X is part of the reality it seeks to account for. Moreover the extra material could in principle allow Y to account for a number of phenomena that are never actually observed, which X itself could not give an account of. Hence Y is less successful in giving an account of why the world is as we observe, since it allows for possibilities that are never realized—a fact which it must remain silent about, but which X immediately explains.
Now assume that we have the same X and Y, and suppose that it's
Y that corresponds to that external reality. In that case, X will be fail to provide an account for those aspects of reality which correspond to the components of Y which are not part of X. Hence we will be able to jettison X, the false hypothesis, as soon as we accumulate sufficient data.
The difference in the two situations is that a more complex false hypothesis incorporating a correct simpler hypothesis can never be falsified, whereas a simpler false hypothesis which is part of a more complex correct hypothesis can be straightforwardly falsified. Hence, methodologically, the only way to arrive at the correct hypothesis is to start from the simplest theory that accounts for the available data and increase the range of observation until one encounters a contraindication. This may or may not actually suffice to falsify the simpler hypothesis, but if it does, then the next most simple hypothesis—the one only more complex enough than the simplest one to account for the problematic data—achieves the priviliged status accorded by Ockham's Razor. And so on.
Application of this approach in a variety of sciences has repeatedly shown that earlier, `gold-standard' hypotheses and the theories that provide interpretations for these hypotheses are not actually wrong, but rather limiting cases of the hypotheses and theories which supplant them in the face of novel, often baffling phenomena. The array of independent domains of evidence which are correctly predicted by modern physical theories identifies these theories as the gold standard of our understanding of that vast reality beyond our senses which we will never have direct access to, but which we can capture abstract models of by the exercise of observation and reason.
