It was a problem for me too when I started learning conformal differential geometry in 2008. As I prefer to learn from openly available sources even though our University has an excellent library within a minute of walk, some of my references will be links to such online resources.
My first encounter with the proof of the conformal transformation of the Christoffel symbols occurred in Jan Slovak's dissertation "Natural Operators on Conformal manifolds", see here.
Later I found a nice exercise (Problem 6.11.8 on p. 296 in Springer 2009 edition) with a solution in P.M. Gadea and J. Munos Masqué "Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers", not freely available but highly recommended :-) This explained a lot to me, however the other things were still difficult to master.
In an old book "Conformal Geometry" edited by R.Kulkarni and U.Pinkall, Vieweg 1988 Bonn, I found a lot of illuminating facts and examples, including these formulas. Especially see J.Lafontaine "Conformal Geometry from the Riemannian Viewpoint", pp. 65-92.
Later I found that there are many texts where one can find all sort of proofs of these identities. Professionals deem them elementary and usually refer to A.Besse's "Einstein manifolds", Theorem 1.159, p.58.
If I were asked now what to read in order to learn these transformation formulas I would recommend Jeff Viaclovsky's lectures "Math 865, Topics in Riemannian Geometry" that you can find here. Lecture 20 (= Chapter 21) on p.78 there gives detailed calculations.
(Reading Spivak's "Comprehensive..." albeit extremely useful was a daunting endeavor for me.
Remark. This sort of questions have been asked already a few times (not reference requests but directly), so I gave some answers, see here and here.