The idea is to give two ways of counting the same thing, one that would yield the left side of the equation and the other the right side of the equation.

For example, on 6.22 we can think of choosing two disjoint groups of size $k$ and $j$ out of $n$ objects. As we can see on the left side of the equation, one way of counting is first choosing $k$ of the $n$ objects, and then choose $j$ of the remaining $n-k$ objects.