The Orsay notes have already been translated, but you want other references.
First read Milnor notes on external rays and local connectivity:
J. Milnor, "Periodic orbits, externals rays and the Mandelbrot set: an expository account," in Géométrie complexe et systèmes dynamiques: Colloque en l’honneur d’Adrien Douady, Paris: Soc. Mat. de France, 2000, pp. 277-333.
J. Milnor, "Local connectivity of Julia sets: expository lectures," in The Mandelbrot Set, Theme and Variations, Lei, T., Ed., Cambridge: Cambridge Univ. Press, 2000, pp. 67-116.
External rays are used to construct puzzle pieces. The pieces around the critical point 0 are nested (each one strictly containing the next. The goal is to compute the annuli between consecutive pieces and hopefully show their sum diverges. If so, the diameter of the pieces around 0 will shrink, and this gives you a system of neighborhoods that prove local connectivity of the Julia set at 0. A related construction gives the same result on the Mandelbrot set M. The details are in Hubbard's explanation of Yoccoz's proof of MLC for finitely renormalizable quadratic polynomials:
J. H. Hubbard, "Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz," in Topological Methods in Modern Mathematics, Houston, TX: Publish or Perish, 1993, pp. 467-511.
After understanding the mechanics of the puzzle construction, learn about Lyubich's principal nest. This is a subset of puzzle pieces with more complicated combinatorics:
M. Lyubich, "Dynamics of quadratic polynomials. I, II," Acta Math., vol. 178, iss. 2, pp. 185-247, 247, 1997.
This allows Lyubich to prove local connectivity of M at infinitely renormalizable pararameters of bounded type:
M. Lyubich, "Dynamics of quadratic polynomials. III: parapuzzle and SBR measures," Asterisque volume in honor of Adrien Douady's 60th birthday ``G\'eom\'etrie complexe et syst\'emes dynamiques'', v. 261 (2000), 173 - 200.
Then you are ready to read the most recent developments:
J. Kahn and M. Lyubich, "The quasi-additivity law in conformal geometry," Ann. of Math., vol. 169, iss. 2, pp. 561-593, 2009.