I found some good explanations of what the Box Cox transform for anyone who
happens upon this thread while searching for boxcox details:

This is the best explanation I found of what how the Box Cox transformation
is calculated.  It explains why QR decomposition is used, has nice code
examples that are probably more illustrative than anything actually
implemented in MASS, which I expect would have a lot of extra mechanics to
make it robust.
http://www.r-bloggers.com/on-box-cox-transform-in-regression-models/
http://freakonometrics.blog.free.fr/index.php?post/2012/11/13/On-Box-Cox-transform-in-regression-models

As usual, Julian Faraway provides some great basic notes on using
tranformations, including the Box Cox transformation:
http://cran.r-project.org/doc/contrib/Faraway-PRA.pdf

A useful, compact, and complete explanation of how the log likelihood
function is derived:
http://data.princeton.edu/wws509/notes/c2s10.html

An useful discussion on the practical limitations of the Box Cox
transformation, and some alternative approaches, such as mixed models.
http://robjhyndman.com/hyndsight/transformations/

A much more complete and helpful help for NextMethod can be found in 5.5 of
R's Language documentation. (Although I'm still not sure how to see the
code in boxcox!)
http://cran.r-project.org/doc/manuals/R-lang.pdf

Also, a more simple and candid summary of S3 and S4 methods is here
http://cran.r-project.org/doc/contrib/R_language.pdf
http://www.r-project.org/conferences/useR-2004/Keynotes/Leisch.pdf

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