In Java, you have code as string. That is, your code is represented using a big string. In Lisp, your code is represented using the list data structure.
So its really code-as-data-structure.
A string is difficult to parse and modify, inserting things in the middle, removing elements, changing the order of the words, that's all really difficult with a string. So if you want to transform Java code, its going to be hard and error prone.
A list is easy to manipulate in contrast. Inserting elements in the middle is trivial, so are deletes and swaps. So in Lisp, if you want to transform code its pretty easy.
Meta-programming is when you write a program that writes a program. An example is say you want to you wanted to add a semi-colon at the end of all your lines of code. You need a macro to do it for you. A macro is a program that acts upon your code to transform it. Eclipse has them. So now its really easy to add a semi-colon at the end of each line. What if you wanted to add a comma between all words in a selection? Now its trickier if your macro operates over a big string, you might need a regex for example. This is meta programming though. Instead of adding the commas yourself, which are required for the program to run, you write another program to add them for you.
Now if all the words were elements in a list, that macro would be a lot easier to write.
This is in essence what code-as-data(structure) means. Its in contrast with code-as-string. You don't have to choose lists as your datastructure either, as long as its something that allows you to represent a turing complete program and is easy to manipulate.
Now, homoiconicity is the fact that your text of code looks like a data-structure too. Making it trivial to parse it into one. So back to my example, you could parse the java code string into a list, and then add commas, and then convert it back to a string. But Java code doesn't map logically into a list. Some construct don't nest like lists, and how do you define what goes into each node of the list? Do you group public and String together? In Lisp, the syntax is an unambiguous AST already, its thus trivial to parse into a list of lists. So you can easily get that AST you need to easily add commas where it make sense.
Finally, there's a third aspect. Code-as-data also implies that your language can accept code as argument in the form of raw data. The best way to think of it is, how would you send a function over the wire to a program and have that program run the code I sent? You need a way to serialize that function, which is code, into raw data that can be transmitted over the wire. The receiving program doesn't have that function defined, so it needs to know how to deserialize it, but also at runtime it must be able to take this raw data, which represents code, and be able to parse it, compile/interpet it and execute it.
Think of SQL, SQL is often used in a code-as-data way. You want "select * from %s". Now you'd take this as a string, and you'd use a string replace, and replace %s with something the user picked in a drop down. At runtime, you are dynamically creating the SQL code, and once you have it, running it. You might have methods that accept SQL and return SQL. Now again, your SQL is a big string, which isn't ideal. But this is still an example of code as data. Now in Java, you can not do that with Java code. Java does not have this concept of code-as-data. In other words, there's no eval.
So when you combine all these three aspects, a homoiconic syntax that parses easily and logically into a data-structure which is easy to manipulate, and where you can then execute data which represent code at runtime and pass it around to other functions, even over the wire, you get a very powerful combo that turns into a Meta-programming powerhouse. This is the strength of Lisps, the one strength all Lisps share.