In my experience people are talking about a couple of different things when saying Lisp enlightened them:

1. The realization that code and data are the same kind of entity. From this realization springs the realization that you can use data structures for managing code and that leads you to Lisp macros - which is one of the superpowers of Lisp. Lisp stands for LISt Processor and Lists are the primary data structure (okay, technically CONSes are, but let's not confuse things) for storing both data and code. Why all the funny parenthesis in Lisp? Because that's the syntax for a list data structure! '(1 2 3) is a list of the first three positive integers and (add 1 2 3) is a list whose first member is the symbol 'add' - which is a function. When evaluated it will yield the value 6. There - now you know Lisp syntax!

2. The realization that many algorithms can be easily implemented using a list data structure having two simple operators: first and rest (CAR and CDR in Lisp parlance) and recursion. This is why you see so many articles about implementing Lisp in <insert popular language here>. It doesn't take a lot to get a Lisp interpreter working, in fact it used to be a routine exercise for undergraduate CS students. Evaluating (car '(1 2 3)) yields 1, and (cdr '(1 2 3)) yields '(2 3). A recursive function in Lisp then typically has two parameters, a list and an accumulator. If the list is NIL then you return the accumulator, otherwise you update the accumulator by applying some operation to the CAR of the list and the input accumulator and recursively invoke the function with the CDR of the list and the updated accumulator. Simple. Tail-call optimization also makes this highly-performant.

3. The difference between symbols in Lisp and variables in other programming languages. In many commonly-used programming languages variables refer to memory locations. That's not strictly true in Lisp. Symbols are an item in a symbol table and the current symbol table is context-sensitive. That means symbols have properties, including user-defined properties, and the same code may be executed with different symbol tables in effect - which is another part of how macros can be so powerful.

4. Functions are first-class objects. This is also different from many other commonly-used programming languages, though that has been changing. A symbol can evaluate to a function rather than a value, functions can be created on the fly (lambdas) and passed as parameters to other functions. You can have a list containing functions (such as (add 1 2 3) from above) and evaluate those functions and so forth. A function is simply another data type. This was more of a big deal 10 or more years ago, now many languages have been adding this capability.

5. The REPL. It's because code and data are the same construct that the REPL as you know it can exist. Even better is when you realize not all functions in the REPL have to be interpreted - you can compile them! You can freely intermix interpreted and compiled code together. You can load compiled code into your session, add new code, modify existing code - whatever you need to do. It's an extremely creative and productive development environment.

That's it! That's the Lisp enlightenment.