The reasoning for including it in US law is discussed in many modern cases. To use one example (Green v The United States 355 US 184 (1957))
"The State with all its resources and power should not be allowed to make repeated attempts to convict an individual for an alleged offense, thereby subjecting him to embarrassment, expense and ordeal and compelling him to live in a continuing state of anxiety and insecurity, as well as enhancing the possibility that even though innocent he may be found guilty"
However its path back through history is much older. In fact - we know that it was part of Athenian law in about 355 BC or so. That's because when Demosthenes was defending 'the young son of Chabrias', he mentioned as part of his argument:
"Now the laws forbid the same man to be tried twice on the same issue, be it a civil action, a scrutiny, a contested claim or anything else of the sort" (You can see this in 'Against Leptines' [Line 147])
It even survived into the Roman era - Iulius Paulus stated that 'after a public acquittal a defendant can again be prosecuted by his informer within thirty days, but after that time this cannot be done' . (Ref: The Opinions of Paulus - Title XVII And Justinian I's collection of Roman Law which was published in 533 A. D ("The Digest of Justinian") stated that 'The governor must not allow a man to be charged with the same offense of which he has already been acquitted'.
This then moved into Canon Law (the Catholic church) under Pope Gregory IX in 1234. This proclamation was "An accusation cannot be repeated with respect to those crimes of which the accused has been resolved". Being a church, however, it uses the Bible to justify this rather than existing laws - in particular St Jerome's interpretation of the Bible verse (written about 7 BC) that 'affliction shall not rise up the second time' Nahum 1:9
This then shifted into English law.
So why did the fledgling USA adapt that particular law?
We know that the early colonists were very familiar with Blackstone's Commentaries on English Law, because they often quoted it to support their claims that the British Parliament was exceeding its lawful authority.
However the colony were also very familiar with the problems the strict English version of double jeopardy caused. Not only was the English version of 'double jeopardy' only limited to capital offenses. Even then it wasn't much help because it was too easy for the crown to simply abandon the case when it was clear that they were losing and then resume it later.
So it was made much more general - in 1641 the Puritan Nathanial Ward published the Colony's first legal code : "No man shall be twise senteced by Civill Justice for one and the same Crime, offence or Trespasse" Massachusetts Body of Liberties #42
However that definition had a problem - strictly speaking it could prevent someone from appealing a court and getting a lighter sentence. So during the constitutional convention James Madison used a slightly different definition of "No person shall be subject ... to more than one punishment .." and after more lawyering it became the Fifth Amendment as:
"No person shall ...be subject for the same offence to be twice put in jeopardy of life or limb;
That was cleverly written - as it doesn't prevent someone asking to be retried years later to be found innocent - simply that they can't be 'put in jeopardy' because of the retrial.
If you want more (including extra references) I've relied largely on :
It's a fascinating area,
-- Mac