Why do criminals who plead guilty receive lighter sentences?

Because it saves the tax payers a huge financial burden and the court prosecutors a lot of work trying to prove they are.

Its a cost cutting deal. They plead guilty, states don't have to spend the money for a trial that can take weeks or months.

For a number of reasons. First because admitting guilt shows character. You show the court you acknowledge that you made a mistake and are willing to accept the consequences. Second because you are saving the court time and money by not going to trial. Third because you are showing respect for the court by not lying to the judge.

Saves the legal system & the publics time & money & the criminal is to have realized the error of their ways.

The DA wants to make sure that he gets punished. In order to know for sure that he will not walk scott free, they are willing to take a lighter sentence. This also saves a lot of money.

Well, it's sometimes part of the bargain for saving the taxpayers money. Trials are expensive and a guilty plea sometimes saves that expense.

This is usually the result of a plea bargain:

"A plea bargain (also plea agreement, plea deal or copping a plea) is an agreement in a criminal case where by the prosecutor offers the defendant the opportunity to plead guilty, usually to a lesser charge or to the original criminal charge with a recommendation of a lighter than the maximum sentence.

A plea bargain gives criminal defendants the opportunity to avoid sitting through a trial risking conviction on the original more serious charge. For example, a criminal defendant charged with a felony theft charge, the conviction of which would require imprisonment in state prison, may be offered the opportunity to plead guilty to a misdemeanor theft charge, which may not carry jail time."
Source and further information:
http://en.wikipedia.org/wiki/Plea_bargain