The Mellin Transform is used to obtain conditions for the validity of Ramanujan's Master Formula/Theorem (RMF). I think reflecting backwards from RMF to the Mellin Transform gives a somewhat intuitive handle on the functionality of the transform, as discussed in the introduction of "[Ramanujan's Master Theorem][1] ..." by Olafsson and Pasquale. I became interested in the Mellin Transform some years ago after reading Hardy's *Ramanujan: twelve lectures on subjects suggested by his life and work*. See also references in Mathworld on [RMT][2].

Update: A simple way to derive the formulas in your question is by looking at the inverse Mellin transform representation of the Dirac delta function. See my short note on the [Inverse Mellin Transform and the Dirac Delta Function][3]. See also some applications in *[Dirac's Delta Function and Riemann's Jump Function J(x) for the Primes][4]* and *[The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions][5]*.

Edwards in *Riemann's Zeta Function in Ch. 10 Fourier Analysis Sec. 10.1 Invariant Operators on R+ and Their Transforms* gives a nice, more group-theoretic intro to the Mellin transform in line with the other comments in this stream.

Note that whereas $e^{sz}$ is an eigenfunction for $d/dz$ and so the Laplace/Fourier transforms are appropriate for devising an operator calculus for $f(d/dz)$, $z^s$ is an eigenfunction of $zd/dz$ and so the Mellin transform is more appropriate for $f(xd/dx)$ . 

  [1]: http://arxiv.org/PS_cache/arxiv/pdf/1103/1103.5126v1.pdf
  [2]: http://mathworld.wolfram.com/RamanujansMasterTheorem.html
  [3]: http://tcjpn.wordpress.com/
  [4]: http://tcjpn.wordpress.com/
  [5]: http://tcjpn.wordpress.com/