Although the origin of hiccups is unknown, the clinical perspective has been fairly well documented. For example, the website of the National Health Service in the UK includes a list of possibly causes of short-term and long-term (chronic) hiccups in humans, as well as a list of medications that have been used in persistent cases. 

For the evolutionary perspective, a great summary of human ancestry can be found in Neal Shubin's 2008 book, Your Inner Fish: A journey into the 3.5-Billion-Year History of the Human Body. This account has also appeared as a PBS documentary. Take a look at the accompanying website for lots of resources and information about your inner fish. A number of brief summaries of the hypothesis connecting hiccups and early vertebrate ancestry have also appeared online including Why We Hiccup and Why Do We Hiccup?

But for a detailed scientific explanation of the two hypotheses about the origin of hiccups presented in this video, we turn to the original literature. The group that has done extensive exploration of the 'phylogenetic hypothesis' is that of Christian Straus and colleagues in Alberta. A link to their article’s abstract is here.

The later hypothesis regarding the possibility that the hiccup is now used as a 'glorified burp' in mammals was best explained by Daniel Howes in this article: Hiccups: A new explanation for the mysterious reflex.