Dyadic dissection with major arcs

In Vaughan's paper "A variance for k-free numbers in arithmetic progressions" https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0024611505015352 he uses at one point a kind of dyadic decomposition for major arcs.

From my understanding, I think he has $$\sum _{q\leq R}\int_{|\beta |\leq 1/qR}|f(\beta )|\hspace {1mm}d\beta $$ and says you can bound this by soemthing like $$\max _{U\leq R}\sum _{q\leq U}\int _{|\beta |\sim U/qR^2}|f(\beta )|\hspace {1mm}d\beta .$$

(His major arcs actually go only to $x/R$ and the second integral has $U/x$ not $U/R^2$, but I think the principle of purely this step is the same, and I'd prefer to have one variable $R$ instead of two $x,R$).

I can follow his proof, but I don't see how to "think" of it, and I feel like there's something I'm missing when I read it. Does anyone have any comments about this step that make it more clear how it works?