This is too thin an answer really, but it's too long for a comment - I expect there'll be much better answers soon enough but hopefully this is at least a start:
If you have any sum, you wonder what you can do to it. (That's maths right?:D) Whenever you have a sum you can do Fourier analysis on, you can try applying Fourier's Inversion Theorem - for the primes, this is all Mellin inversion, i.e. Perron's formula. For divisor functions, it's Voronoi's summation formula. In each case you started with a sum and now have a new sum - the dual sum.
Maybe the new sum is better, maybe it's not. If it is better and you get new results, then later people can look at it more philosophically/deeply and try to figure out why it worked, what really goes on, etc., probably in the hope of generalising it to other situations, but in practice the reason why it was attempted in the first place was maybe just to see what happens.
So you consider the Dirichlet series just to see what you can say about the new sum after applying an inversion theorem - a contour integral for anything you're looking at the Dirichlet series of. Of course, it was Riemann's insight to realise you can move the contour by extending $\zeta (s)$.
By the way, have a look at Chapter 5 of Koukoulopoulos's book - there are many good books of course but (imo) this one doesn't shy away from really explaining the why's.