For example, given the following:
Minimise $$ f(x_1, x_2) $$
Subject to $$ h(x_1, x_2) = 0 $$
$$ g(x_1, x_2) \leq 0 $$
The KKT conditions are written out as
$$ l(x, \lambda, \mu) = f(x_1, x_2) + \lambda h(x_1, x_2) + \mu g(x_1, x_2) $$
And one of the conditions is that
$$ \mu \cdot g(x_1, x_2) = 0 $$
I'm not sure why this condition is necessary though, I think that it has something to do with the constraint being active / inactive and/or there being slack? I'm not sure what the reasoning is though.
The condition $\mu \cdot g(x_1,x_2) = 0$ is a complementary slackness condition. It says that either the dual multiplier $\mu$ is 0, or there is no slack in the $g(x_1,x_2) \le 0$ constraint.
The dual values give the change in the optimal objective function value if the right-hand side of the constraint changes. So the intuition behind the complementary slackness condition is that if there is slack in the constraint ($g(x_1,x_2) \lneqq 0$), then changing the RHS would have no effect on the optimal objective function value, hence $\mu=0$. And, if $\mu \ne 0$, then a change in RHS would lead to a change in the objective function, so the constraint must be tight ($g(x_1,x_2) = 0$).