A practical application is for example in probability theory (or anywhere that deals with measures) such as this question:

If we generate a random real number from 0 to 1, what is the probability that it is rational?

Because we know that the continuum is so much larger in a sense than the set of rationals, we can answer this confidently and say the probability is zero, even though it is theoretically possible for us to get a rational number.

Statistics deals with similar scenarios quite frequently, and without it we wouldn't have the modern scientific method.

Practical application in math tends to be like three degrees of separation and half a century removed from the math at play. In this case, all of modern mathematics is based on set theory, so it's more that this stuff allows us to do other, more practically useful math while knowing what we're talking about.

I've never been a fan of just saying "some infinities are bigger than others," to be honest. Way too easy to misunderstand and it's also kind of meaningless by itself.

The real numbers also includes the integers.

The practical consequence of this example is that the integers die regardless of what you choose.

However infinitely many people will survive if you choose the first option.