The condition people drop is the important one. The razor applies when two explanations account for the evidence equally well. If one is simpler but explains less, the razor says nothing in its favour — it is simply the weaker theory. Nearly every bad use of Occam's razor consists of forgetting this and treating "simpler" as sufficient on its own.
William of Ockham, a fourteenth-century Franciscan, did not write the phrase most associated with him, and the underlying idea is older — Aristotle held that nature does nothing in vain, and comparable principles of parsimony appear throughout medieval philosophy. Ockham's own applications were largely to metaphysical entities: he argued against postulating universals as things existing over and above particulars, on the grounds that they did no work you could not do otherwise.
Why should parsimony count at all? The best justification is not that the universe is simple — there is no guarantee of that — but that unnecessary assumptions are unsupported ones. Every additional entity you postulate is a further claim about the world that your evidence does not require, and therefore a further way your theory can be false. A theory with fewer independent commitments has fewer opportunities to go wrong, which is a reason to prefer it that does not depend on any assumption about nature's tidiness.
It has done real work in the history of science. Copernicus's model was preferred partly on grounds of economy over the accumulated epicycles of the Ptolemaic system. Lavoisier's chemistry eliminated phlogiston, a substance postulated to explain combustion and doing no work once oxidation was understood. In both cases the simpler theory was also at least as good empirically, which is exactly the condition the razor requires.
LimitsWhat the razor cannot do.
Simplicity is hard to measure. Simpler in what respect — fewer entities, fewer kinds of entity, fewer equations, shorter description? Theories can be simpler on one measure and more complex on another, and the razor gives no way to decide which measure applies.
Reality is sometimes complicated. Quantum mechanics is not simple; neither is evolutionary biology. The razor never promised the world would be tidy, only that we should not add assumptions our evidence does not support.
It's a guideline, not a proof. Preferring the simpler theory does not establish it. The razor tells you where to place your provisional confidence pending further evidence, which is a much weaker claim than it is usually made to carry in argument.
The most common misuse in practice is deploying it as a conversation-ender — declaring one's own position simpler and treating the matter as settled. Since the razor only operates between explanations of equal adequacy, that move nearly always begs the question it claims to answer.