Reference for an undecidability proof
to some degree all refs on the subject have a formidable inherent complexity due to the subject matter. however, here is a classic, highly accessible ref on Gödel's thm that may be helpful, written in "popular science" style by a philosopher &amp; mathematics historian for a broad audience, breaking it down into intuitive everyday language.


Gödel's Proof Nagel/Newman


here are some other similar but more technical refs that may be helpful


An Introduction to Gödel's Theorems Smith
Gödel's Theorem: An Incomplete Guide to Its Use and Abuse Franzén


there are also many refs on Turing's halting problem. one worth mentioning, a step-by-step walkthrough/tour of the original proof, but which is very advanced due to retaining all the mathematical complexity/detail: 


The Annotated Turing: A Guided Tour Through Alan Turing's Historic Paper on Computability and the Turing Machine Petzold


on the other hand most of the proof complexity of the halting problem is associated with the proof of the existence of the Universal TM, wikipedia has a pretty good outline. this can be understood as a simulation or emulation of one machine by another. therefore studying the theory of machine emulation (now quite frequently applied with video games) may be helpful.