This article explains the function of Liberty Bells in Sid Meier's Colonization.
The goal of the game is to achieve independence from Europe. It is, however, impossible to declare independence unless at least 50% of the population supports it. So how do you measure how many people support independence, and how do you increase that number?
Liberty Bells, Sons of Liberty, and Rebel Sentiment:
Liberty Bells are produced at the Town Hall colony building. The Town Hall automatically produces one liberty bell, and by assigning colonists to work there it will produce more bells. The specialist for the Town Hall building is the Elder Statesman. Building a Printing Press and later a Newspaper will increase the number of liberty bells produced at the Town Hall. Liberty bells increase Sons of Liberty membership within a colony over time, and subsequently increase Rebel Sentiment. Liberty Bells also cause Founding Fathers to arise within your population.
(Images: Town Hall, Town Hall with Elder Statesman, Printing Press, Newspaper)
The percentage of the total population supporting independence from Europe is called Rebel Sentiment. So if the rebel sentiment is 30%, that means 30% of the total population supports the idea of independence.
The Sons of Liberty memberhip tells you how many people within a colony support independence. This number can be higher or lower than the rebel sentiment. In the image below (taken from the colony screen) you'll notice that five of the colonists (60%) support the king, represented by the blue crown, and three colonists (40%) support independence. Those who still support the king are called Tories. (If you're curious about how Sons of Liberty membership is calculated, you might want to check out this forum thread.)
In short: Sons of Liberty membership and Rebel Sentiment are measurements of how many people support independence, but on different levels (colony level vs. total population), and these numbers increase over time depending on how many Liberty Bells you produce.
Importance of Liberty Bells production:
As stated previously, Liberty Bells production is important because the ultimate goal of Colonization is to achieve independence from Europe, and a majority of the population has to support that endevour. There are, however, many other reasons as well:
Production penalty/bonus: If the number of Tories (those who support the king) in a colony reaches a certain number, that colony will get a production penalty of -1 per colonist. At the easiest difficulty level, Discoverer, the penalty hits you at 10 Tories, and at Viceroy it hits you at 6 Tories. On the other hand, if Sons of Liberty membership reaches at least 50%, all colonists get a production bonus of +1. At 100% Sons of Liberty membership, the production bonus is +2 per colonist, making that colony extremely productive. Colonists are also educated faster in a colony with high Sons of Liberty memberhip.
Founding Fathers: Liberty Bells production causes Founding Fathers to appear in your Continental Congress. The more bells you produce, the faster these assets appear. Some Founding Fathers in turn increase liberty bell production and rebel sentiment: Thomas Jefferson, Thomas Paine, and Simon Bolivar.
Declaration of Independence: Additional benefits of Liberty Bells production come into play when you declare independence from the king; these being units upgrade, an intervention force, and finally additional points added to your total Colonization score.
As you can see, the production of Liberty Bells is very important from beginning to end in Sid Meier's Colonization.