What does this phrase about economics mean [Lionel Robbins]?
To me, the quote seems to be about what economics is, and, more subtly, what it isn't. Robbins claims economics is about scarce resources that people can use for multiple things. The implication, and you will find this in undergraduate economics textbooks, is that economics is concerned with efficiency. Economists are simply looking for ways to make the most out of scarce resources. An outflow of this focus on efficiency is the paradigm of utility or profit optimization under constraints.

Perhaps more interesting is the subtle claim of what economics is not about: the distribution of resources. This does not seem particularly radical now, but it was much more so at the time. Keynes and Marx still dominated the political discourse in the early 20th century, and both were very much concerned with the distribution of resources. By focusing on efficiency rather than distribution, Robbins puts himself firmly in the neoliberal paradigm.