Motivation for blowing-up

Question

Let $X =\operatorname{Spec}(A)$ and $Z \to X$ is a closed subscheme given by the ideal $I$. The blowup along $I$ is defined to be $\operatorname{Proj}(A \oplus I \oplus I^2 \oplus\cdots)$ which seems a bit unmotivated to me. I can follow proofs in Hartshorne, but I do not see how one can come up with such a definition. Is there any intuitive approach to it?