I'm a novice in Lisp (specifically Clojure) and I got interested in the language because of the macro system.
However, all the examples I've seen so far seem pretty convoluted and just made me think "Why would I want to do that?". I think I just didn't "get" it yet.
Is there a simple real-life example that shows the usefulness of the Lisp macro system to a beginner?
No; all the simple macro examples are in fact in the "why would I do that" category. Their value is the how, rather than the why.
A very simple macro can often be eliminated by writing whatever it writes by hand without a lot of additional effort, and without the baggage of carrying the macro definition. It doesn't seem useful, especially if you fail to imagine it being used thousands of times, rather than just the two or three times that occur in the example.
It's the convoluted macros that answer the "why": because if you don't have the convoluted macro to generate code in a complicated way, and you need to do that dozens of times, then your alternative is to do it all by hand, which is tedious and error-prone. THe macro does the job consistently, and if there is a bug, you just fix it in the macro and re-expand all the uses of it to spread the bugfix.