The specific numbers in the question are from CCIR 601 (see the Wikipedia link below).
If you convert RGB -> grayscale with slightly different numbers / different methods, you won't see much difference at all on a normal computer screen under normal lighting conditions -- try it.
Here are some more links on color in general:
Wikipedia Luma
Bruce Lindbloom 's outstanding web site
chapter 4 on Color in the book by Colin Ware, "Information Visualization", isbn 1-55860-819-2; this long link to Ware in books.google.com may or may not work
cambridgeincolor : excellent, well-written "tutorials on how to acquire, interpret and process digital photographs using a visually-oriented approach that emphasizes concept over procedure"
Should you run into "linear" vs "nonlinear" RGB, here's part of an old note to myself on this. Repeat, in practice you won't see much difference.
In color science, the common RGB values, as in html rgb( 10%, 20%, 30% ), are called "nonlinear" or Gamma corrected. "Linear" values are defined as
Rlin = R^gamma, Glin = G^gamma, Blin = B^gamma
where gamma is 2.2 for many PCs. The usual R G B are sometimes written as R' G' B' (R' = Rlin ^ (1/gamma)) (purists tongue-click) but here I'll drop the '.
Brightness on a CRT display is proportional to RGBlin = RGB ^ gamma, so 50% gray on a CRT is quite dark: .5 ^ 2.2 = 22% of maximum brightness. (LCD displays are more complex; furthermore, some graphics cards compensate for gamma.)
To get the measure of lightness called L* from RGB,
first divide R G B by 255, and compute
Y = .2126 * R^gamma + .7152 * G^gamma + .0722 * B^gamma
This is Y in XYZ color space; it is a measure of color "luminance".
(The real formulas are not exactly x^gamma, but close;
stick with x^gamma for a first pass.)
Finally,
L* = 116 * Y ^ 1/3 - 16"... aspires to perceptual uniformity [and] closely matches human perception of lightness." -- Wikipedia Lab color space