p.s. I already know few things, but it is interesting to listen to different perspectives.
what I know: the augmented Lagrange method combines the penalty method and the method of Lagrange multipliers. The penalty method convergence hinges heavily on the penalty parameter which must tend to $\infty$ to approach the optimal solution. By adding another quadratic constraint, the ALM method relaxes this and hence a finite value of the penalty parameter suffices.