There is no exact account of the alleged loophole from the time (nobody wrote it down), but many attempts at reconstructing it, or similar loopholes, have been made. This article (warning: automatically downloads a PDF) discusses some of them, the background, and what we do and don't know about it (and puts forward a long argument about what the "loophole" may have been). The basic gist of all of them is that the Constitution provides a means for amending itself, even amending the requirements for amendment, which means that any freedoms promised by it can be undone if enough people get on board. Which is not especially profound (no more profound than noting that a President can, and has in the past, temporarily suspended parts of the Constitution, and that his Commander in Chief powers are very broad; or that the US Supreme Court has at times allowed for vast injustices to take place; or that much of the operation of our existing republic — including judicial review — is a matter of traditions and is not strictly codified in the Constitution, and thus fragile, etc.).

The argument that this is perhaps Gödel's loophole is based on his own interests in logic and incompleteness — if you have a Constitution that can amend itself, then ultimately it could amend itself to any end, including dictatorship. True enough. What makes this uninteresting from a political/historical perspective is that Constitutional republics are not that robust anyway, and most have historically just been upset by coups and other blunt, violent power grabs, so appealing to unsolvable logical paradoxes is a pretty round-about way of suggesting what could be rather obvious from the study of history or political realities.

Furthermore I suspect literally no one who has actually read the Constitution, much less studied Constitutional law or practice, has ever suggested that the Constitution is a logically complete and coherent document. It is an 18th-century creation, by 18th-century men, and has been modified and amended over the centuries, and bears the historical mark of fallible humans and political compromises throughout its core. Gödel might have been shocked or amused to discover logical paradoxes in it, but the reality is that a historian would have told you, much more quickly and with much more compelling evidence, that of course the United States could descend into dictatorship despite the Constitution. It is of note that when the US has exported its particular model, it has tended to result in coups, at higher rates than governments with a Parliamentary model, where the executive is much weaker relative to the legislative branch. This is only to say that it doesn't take an world-class logician to suggest there might be reasons to think that American democracy is not as institutionally robust as it is sometimes claimed to be.

To add to what u/restricteddata said, I would mention that the Constitution is neither a long nor a particularly complex document, allowing for 18th-century language issues. It's about 4,500 words in the original, and if you include the amendments you get to about 7,600 words.

Constitutional law and the interpretation of the Constitution is complex, of course, partly due to the fact that the Constitution was written to an extent to be intentionally vague.

Anyhow, if you can set aside half an hour or so you can easily read and digest the document and the amendments, confusingly linked by the National Archives in two sections, for yourself.

(I don't know how to add the two dots above the 'o'. sorry)

You can get around that by simply adding an e after the letter. Goedel. Works for all German names/words that have ä, ö, or ü. The letter ß can be replaced by ss.