For starters, it's more of a tradition than an actual rule. The 2 term limit dates back to the first President, George Washington. Given the time he was in office, Washington was a surprisingly humble figure (refusing the title of "His Excellency" in favor of "Mr. President" and actually believed in the government he fought for. Towards the end of his first term, he was quoted as saying "I did not fight George the Third so I could become George the First" when questioned why he did not seek a third term
Historically, the tradition stuck. Presidents normally didn't run for a third term as it was deemed that after 8 years, either out of respect for Washington's dedication to the nature of the office, or because they died in office, or they were just so unpopular they wouldn't have been elected. The first President to buck this trend was Franklin Roosevelt (FDR) who successfully ran for a third term... followed by a fourth term, which he died during the course of.
At this point, there was a significant push to enshrine the office as a two term limit for the sake of changing status quo and in 1951 they drafted an amendment to the constitution doing just that (Though it allowed the incumbent FDR and VP Truman to be exempt from this law until they were voted out, though this did not happen. As mentioned, FDR died in office and Truman succeeded him for the remainder of his term and was re-elected for a full term, and then retired.). At present the maximum amount of time one can be President is 10 years though this has yet to happen (You must be the Vice President, who succeeds the President with less than two years remaining in a term, and then goes on to win two terms of his own).
Again, this isn't so much about democracy as it is tradition... Prior to FDR, Presidents Grant and Theodore Roosevelt (Yes, Teddy is related to FDR) both sought a third term, but were unsuccessful. The fact was that most of the Presidents who stood a chance of winning a third term stuck with a tradition and decided not to run again. For the bulk of the history of the country, it was just the way we were and no one was forced to do it.