What the Black-Scholes PDE is telling you

In words, what does the Black-Scholes partial differential equation express?

That a delta-hedged option earns exactly the risk-free rate. Writing it as $\Theta+\tfrac12\sigma^2S^2\Gamma=r(V-S\Delta)$ shows the balance: the hedged position loses time value (theta) and gains from convexity (gamma) as the stock moves, and the net must equal interest on the cash in the hedge. The real drift $\mu$ is absent, so prices depend only on volatility, rates and the payoff.