Girsanov's theorem and switching to risk neutrality

What does Girsanov's theorem say, and why is it so important for derivatives pricing?

Changing to an equivalent measure shifts the drift of a Brownian motion but leaves its volatility unchanged. If $W$ is a $\mathbb{P}$-Brownian motion and $\mathbb{Q}$ has density $\exp\big(-\int\theta\,dW-\tfrac12\int\theta^2dt\big)$, then $W_t+\int_0^t\theta_s\,ds$ is a $\mathbb{Q}$-Brownian motion. In finance this lets us replace the real-world drift $\mu$ with the risk-free rate, so prices become discounted expectations under the risk-neutral measure, and it underpins numeraire changes and importance sampling.