(EDIT: This answer partially overlaps some comments on math.SE mentioned by Nate, see here.)

The second transformation has no interpretation that I am aware of. Could you provide some context?

The first transformation (from $X$ to $Y$) is called *size-biasing*. It is equivalent to ask that $E(\varphi(Y))=E(X\varphi(X))/E(X)$ for every test function $\varphi$ (for every bounded measurable $\varphi$, if you like). As such it can be defined for every nonzero integrable distribution of $X$ on the positive real halfline, discrete or continuous or otherwise. (Note that if one does not assume that $X\ge 0$ almost surely and that $E(X)$ is positive and finite, your condition makes no sense.) In particular, as noted by Suresh, your $\lambda$ is $E(X)$, but also $E(Y^k)=E(X^{k+1})/E(X)$ for every nonnegative $k$, the Laplace transform of the distribution of $Y$ is related to the first derivative of the Laplace transform of the distribution of $X$, and so on.

There are some beautiful uses of this transformation in the context of branching processes, as explained in a forthcoming book Probability on Trees and Networks by Russell Lyons with Yuval Peres.