I am looking for some good introductory articles or books on "reflective equilibrium method". I am more interested in practical use of this method than in its philosophical foundations etc.
You need to check Nelson Goodman's Fact, Fiction, and Forecast, pp.66-67, which is one of the earliest discussion of it; the idea is however made famous by John Rawls in his A Theory of Justic, pp. 48-51.
Check out W. Van der Burg and T. Van Willenburg (eds.) Reflective Equilibrium: Essays in Honour of Robert Heeger. Dordrecht. Our paper Mäkinen, J. & Kakkuri-Knuuttila, M-L (2013) The Defense of Utilitarianism in Early Rawls: A Study of Methodological Development Utilitas 25(1) studies especially Rawls's Two Concept of Rules (1955) article from the methodological perspective.
A very good introduction into the method of reflective equilibrium is Michael R. dePaul's, "Intuitions in moral inquiry", chapter 21 of The Oxford Handbook of Ethical Theory, ed. by David Copp, OUP, pp. 595-623.