Ratings are calculated using the Glicko-2 rating method developed by Mark Glickman. This is a very popular rating method, and is used by a significant amount of chess organisations (FIDE being a notable counter-example, as they still use the dated Elo rating system).
Fundamentally, Glicko ratings use "confidence intervals" when calculating and representing your rating. When you first start using the site, your rating starts at 1500 ± 700. The 1500 represents your rating, and the 700 represents the confidence interval.
Basically, the system is 90% sure that your rating is somewhere between 800 and 2200. It is incredibly uncertain. Because of this, when a player is just starting out, their rating will change very dramatically, potentially several hundred points at a time. But after several games, the confidence interval will narrow, and the amount of points gained/lost after each game will lessen.
Another thing to note is that the confidence interval changes over time. If you win or lose many games (or rather "points") in a row, the confidence interval will increase allowing you to gain/lose points points more rapidly. This is because a winning/losing streak means that you are incorrectly rated/seeded and the rating system should compensate for that.
"Why use 1500, and why use Glicko in the first place? I much preferred it when the ratings started at 1200 and Elo was used; it was far more accurate with FIDE ratings."
1500 is used because it is the figure specified by the Glicko-2 documentation that can be accessed here: www.glicko.net/glicko/glicko2.pdf
The change from Elo to Glicko was a conscious decision. The main issue with Elo was that ratings moved too rapidly. Against a player of equal strength, it was possible to gain or lose 100 points in less than a handful of games. There were other issues as well, such as most players (of vastly different strengths) being bulked together near 1200. This was incredibly problematic.
After changing to Glicko-2, players have been far more evenly spread-out within the rating ranges, making it far easier to find an opponent of appropriate strength. It is also far easier to maintain a rating.
"But ratings are far higher compared to other sites and organisations such as FIDE, USCF, the ICC, etc... What's up with that?"
It is best not to think of ratings as absolute numbers, or compare them against other organisations. Different organisations have different levels of players, different rating systems (Elo, Glicko, Glicko-2, or a modified version of the aforementioned). These factors can drastically affect the absolute numbers (ratings).
It's best to think of ratings as "relative" figures (as opposed to "absolute" figures). By this I mean: within a pool of players, their relative differences in ratings will help you estimate who will win/draw/loss, and how often. Saying "I have X rating" means nothing unless there are other players to compare that rating to.