As a complement to more scholarly references, you might enjoy [this video recording][1] of a delightful interview with Kirby. In the section labeled "The Torus Trick," he gives a description of the trick which I found enlightening both mathematically as well as historically (though it's a bit brief). Some of the surrounding sections of the video contain further interesting information about the state of topology at the time.


  [1]: https://www.simonsfoundation.org/science_lives_video/robion-kirby/