'Homoiconic' is kind of a vague construct. 'code is data' is a bit clearer.
Anyway, the first sentence on Wikipedia for Homoiconic is not that bad. It says that the language has to have a source representation using its data structures. If we forget 'strings' as source representation (that's trivial and not that helpful to have a useful concept 'homoiconic'), then Lisp has lists, symbols, numbers, strings etc. which are used to represent the source code. The interface of the EVAL function determines what kind of source representation the language is working on. In this case, Lisp, it is not strings. EVAL expects the usual variety of data structures and the evaluation rules of Lisp determine that a string evaluates to itself (and thus will not be interpreted as a program expression, but just string data). A number also evaluates to itself. A list (sin 3.0) is a list of a symbol and a number. The evaluation rules say that this list with a symbol denoting a function as the first object will be evaluated as a function application. There are a few evaluation rules like this for data, special operators, macro applications and function applications. That's it.
To make it clear: in Lisp the function EVAL is defined over Lisp data structures. It expects a data structure, evaluates it according to its evaluation rules and returns a result - again using its data structures.
This matches the definition of homoiconic: source code has a native representation using the data types of Lisp.
Now, the interesting part is this: it does not matter how EVAL is implemented. All that matters is that it accepts the source code using the Lisp data structures, that it executes the code and that it returns a result.
So it is perfectly legal that EVAL uses a compiler.
(EVAL code) = (run (compile-expression code))
That's how several Lisp system work, some don't even have an Interpreter.
So, 'Homoiconic' says that the SOURCE code has a data representation. It does NOT say that at runtime this source code has to be interpreted or that the execution is based on this source code.
If the code is compiled, neither the compiler nor an interpreter is needed at runtime. Those would only be needed if the program wants to eval or compile code at runtime - something that is often not needed.
Lisp also provides a primitive function READ, which translates an external representation (S-Expressions) of data into an internal representation of data (Lisp data). Thus it also can be used to translate an external representation of source code into an internal representation of source code. Lisp does not use a special parser for source code - since code is data, there is only READ.