You can't see a date on it, but that page is now 10 years old (I wrote my entry when I was 20 years old, 30 now). Still loving Common Lisp (and clojure and Elixir and all the many other cool things that came out since then), so no harm done; just... beware (-:
I'm 15 and after trying to learn several different kind of languages(all from BASIC to C) I've finally found one that seems good for a total beginner. Even though SICP is a quite advanced text for someone who ain't native english speaking nor being very good at math, I've found it as an excellent book which I highly recommend to other beginner's.
Whenever I don't understand something I read the last parts again and finally I understand it. I've also got some great help by the CHICKEN community and by reading articles at Wikipedia if there's words and math I don't understand.
Lisp is very good at parsing language (i.e. all data, like computer programs, are read into lists and LISP is built around this functionality; this is one reason why it is very popular among AI researchers). Combine that with things like 'eval' statements and you can easily parse and execute an arbitrary computer language with LISP. Very, very cool stuff.
They probably mean something like macros and embedding DSLs.
What they probably don't mean is any sort of non-trivial automatic programming--writing programs that, for example, search through possible ways to solve some particular programming problem in assembly. Which, amusingly, is the one thing I've been using lisp for lately, working on program synthesis.
Of course, I would probably be a poor addition to this page because I would much rather use Haskell or OCaml. And because I'm using Racket rather than Common Lisp, which looks to be the theme of the page.
Think of a programming language. It has some built-in data types, and it has so-called literals for each of those. The parser reads in some text and produces a value of some kind. In Lisp, the program itself is really one large "code literal", which happens to be run by the language environment. But there are plenty of other things you can do with programs besides just running them, and generating them is one of those things. Lisp merely provides handy facilities for performing these manipulations.