What happens "legally" when you get married?

This question is prompted by this question.

You can either get married in a church or before a judge or justice of the peace. In either circumstance, the minister, judge or JP is licensed by the state to administer an oath to you, and witness your signature, in the same manner a notary will witness your signature on a legal document. In order to get married, you must take out a marriage license, pay certain fees, and in some states have a blood test to prove you’re not marrying your sibling.

Is the marriage license actually a license? If so, what are you licensed to do? Or is it a contract? If so, what dictates its terms?