Robbins's famous quote can be interpreted very differently. For me, all he tried to suggest with this is that economics is the study of how you choose.
Most textbooks would say that if you have something for free, it is not a matter for economics to study. For instance, you can freely sunbathe, so the amount of time you spend sunbathing is not an economic problem. However, Robbins's definition claims that it can be a matter for economics, even if it is free. If you have to choose between sunbathing and going to the movies, the relevant resource you are allocating is not only money but also time.
In sum, given that doing A implies sacrificing doing B instead of A Lionel Robbins is defining economics as the science that studies human (purposeful?) behavior.
Btw: the original text, and, therein:
Scarcity of means to satisfy ends of varying importance is an almost
ubiquitous condition of human behaviour (Robbins, 1945, p. 15)
PS: I don't believe that Robbins is suggesting that one focuses on efficiency as opposed to distribution. If the distribution affects choices, then it is a matter for [Robbins's] economics; if by distribution you are asserting that this distribution of income is better than that one, than it is true that Robbins does claim
What is of relevance to the social sciences is, not whether individual
judgments of value are correct in the ultimate sense of the philosophy
of value, but whether they are made and whether they are essential
links in the chain of causal explanation. (Robbins, 1945, p. 90)
PPS: Reading now, this sounds too pompous, but at least I hope it's helpful:)