Because you choose to stop. Taking one billion steps, each one half the size of the previous step, will get you really close to the finish line, but not quite there. Like you said. But an infinite series of steps isn't one billion or ten billion or one hundred billion, it's infinite. That's why it's a tricky question. It's easy to imagine lots and lots of steps and say that's how an infinite series behaves, but it's not quite so.
Limits as variables go to infinity are no fun, I know. Replacing infinity with a really big number doesn't always work.
So the answer to the question is: an infinite geometric series doesn't sum to infinity, it sums to a finite number: that's why you can cross an infinite number of intervals and still cross a finite distance.
To me, the cat/cardboard mystery is harder to figure out.