First comes a bureaucratic answer.
We have a group of $n$ people. We want to select a committee of $b$ people, and choose $a$ of them to be a steering subcommittee.
The right-hand side counts the number of ways to pick the committee of $b$ people, and then choose the steering subcommittee from this committee.
The left-hand side picks the steering subcommittee first, then the rest of the committee.
Both sides count the same thing, so they are the same.
Or else we want to choose $b$ people from a class of $n$ to go on a trip. Of these $b$ people, $a$ will ride in the limousine, and the rest in an old bus. We can choose the $b$ people, and then choose the $a$ of them who will ride in the limousine.
Or else we can choose the $a$ people who will ride in the limousine , and then pick $b-a$ people from the remaining $n-a$ to ride in the bus.