Great question. Let's let r be a rational number and s be irrational. Note r has to be nonzero for this to work. In other words, it's not true that when we multiply zero, a rational number, by an irrational number like π we get an irrational number. We of course get zero.
The question is: why is the product

irrational?
In math "why" questions are usually answered with an illuminating proof. Here the indirect proof is enlightening.
Suppose p was rational. Then

would be rational as well, being the ratio of two rational numbers, so ultimately the ratio of two integers.
But we're given that s is irrational so we have our contradiction and must conclude our assumption that p is rational is false, that is, we conclude p is irrational.