It may seem strange to define $0\cdot\infty=0$. However, one verifies without difficulty that with this definition the commutative, associative, and distributive laws hold in $[0,\infty]$ without any restriction.
The way this is worded leads naturally to your question, as though Rudin were implying that this is the main justification for defining $0\cdot\infty$ in this way. Rather, I see this as a bonus after making the convention consistent with what happens when integrating the $0$ function or integrating over a space of measure $0$, as KCd's comment indicates.
If you ask yourself what the possibilities are, you can start by supposing that $0\cdot\infty=x$ for some $x\in[0,\infty]$, and apply the distributive law to see that $x=2x$, so that $x=0$ or $x=\infty$. You can then verify that with either convention the commutative, associative, and distributive laws will hold, so something more is needed to motivate the choice. Such a choice will always depend on context, and in some cases it won't be a good idea to even define $0\cdot\infty$. However, I am not aware of a mathematical context in which the convention $0\cdot\infty=\infty$ is useful.