I want to understand the logic for why this is:
We have our model for the stock price behaviour:
$$d{S_t} = \mu {S_t}dt + \sigma {S_t}d{\tilde W_t}$$
It describes the development of a stock price over time using the risk-adjusted expected return $\mu$ and the real uncertainty in the stochastic term. We want to change the probability measure in such a way that the stochastic process remains a Brownian motion but with a drift of r instead of $\mu$. .... To repeat the manner of speaking, we want the process to change gear from an instantaneous increase of $\mu$ to r and leave the rest as before.
$$d{S_t} = r{S_t}dt + \sigma {S_t}d{\tilde W_t}$$