Vaughan's identity https://proofwiki.org/wiki/Vaughan%27s_Identity is a very useful identity in analytic number theory. The identity expresses the von-Mangoldt function $\Lambda(n)$ as a sum of several sums.

I have seen a couple of applications of Vaughan's identity and I can follow the proof fine but I don't seem to have the intuition on why it works or why the identity is how it is. At the moment the identity seems like expressing $\Lambda(n)$ as a sum of random sums to me...(which I am very sure is not the case!). I would greatly appreciate some explanation on why Vaughan's identity is how it is. Thank you very much!