Why is the 1983 quarter worth so much?

It's worth more than other quarters of that period, but it's still not a hugely valuable coins. About 1.3 billion were struck so they're certainly not rare.

Interest in them stems from the fact that among that number, a lot of them were poorly-struck so nice-looking specimens aresomewhat less common. The Mint was having a lot of trouble with dies whose characteristics were better suited to the older, softer silver-copper alloy used up to 1964. The harder cupronickel metal in modern coins was causing dies to deform so that images sometimes appeared to slide off the edge of the coin blank. If you look at some 1982 quarters you'll see that the date and the Word LIBERTY may actually be merged into the coin's rim.

The Mint's first attempts to fix the problem involved reducing the amount of relief in the designs; i.e. the images weren't so high off the flat background. Problem was, the lowered relief was SO low that many coins came out weakly struck and were at least as ugly as their deformed predecessors. It took until the mid to late 80's for them to get everything adjusted correctly.