There's a good elaboration of it in Tom Stuart's "Programming With Nothing" talk[0], in which he does a lambda calculus implementation of FizzBuzz in what's left of Ruby if you leave everything out except the things strictly necessary to do lambda calculus - with the exception of what you could call C-style macros (basically text search-and-replace, so you can call a section of code FOO and just type FOO everywhere you would have typed that section of code). There's also the little matter of output - you need to translate the resulting Church numerals into readable form for those who can't see that you've obviously produced correct FizzBuzz. There's a good enough explanation that what needs to be sort of hand-waved for brevity (things that are shown but not really explained in detail) aren't entirely opaque - you will be able to figure it out.

Keep in mind, too, that the lambda calculus is a lot like machine language in one respect - the same little bit of "code" can have more than one meaning to the programmer even if the "machine" is doing exactly the same thing.

[0] https://www.youtube.com/watch?v=VUhlNx_-wYk