The marriage license is a government issued document that makes the marriage a legally binding contract. From a bureaucratic viewpoint, the marriage license is no different than any other legal contract. It serves as a government record that the two parties listed on the document are contractually obligated to each other, in other words, they are legally married.
Most people don't know this, but the government issued marriage license is a relatively recent cultural development. With the exception of Massachusetts, there was no such thing as a marriage license in the United States until about the 1920's. Even then, it took another few decades for the marriage license to catch on in all 50 States and territories.
Prior to the institution of government issued marriage licenses, marriage was administered and recorded by various communal and religious institutions. This is the main reason why Common Law Marriages exists. It was partially intended to grandfather in marriages that took place before state governments took over the administration of marriage.
The history of marriage licenses in the US is a subject that is way too expansive to cover here. However if you are interested in learning more about the history of marriage licenses, you can check out the following links.