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 ..." 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.
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 the first draft of my note on The Mellin Transform.
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.