Intuitively, why is the complementary slackness condition true?
As you have noted, complementary slackness follows immediately from strong duality, i.e., equality of the primal and dual objective functions at an optimum.
Complementarity slackness can be thought of as a combinatorial optimality condition, where a zero duality gap (equality of the primal and dual objective functions) can be thought of as a numerical optimality condition.

In order to understand what complementary slackness means, the concept of dual variables as "shadow prices" is useful. The dual variable associated with a primal constraint is called the constraint's shadow price because it can be thought of as how much the objective function would increase if the constraint was relaxed (meaning e.g. the right hand side of a $\le$ constraint was increased). 

Complementary slackness says that at an optimal solution, if a shadow price (dual variable) is positive, meaning that the objective function could be increased if the corresponding primal constraint was relaxed, then this primal constraint must be tight. If not, the primal objective function value could be improved (by changing the primal variables in order to make this non-binding primal constraint binding).