Does it make the remote more "accurate," or does it add some features? Either way, why would I want a remote with it over a regular one? Are there games that can only be played with the attachment, or does it just make playing some games easier/better?
The Wii MotionPlus was created because it enables the controller to detect horizontal movement such as thrusting the controller or swinging it horizontally. The original controller could only pick up vertical movement well. It had a single accelerometer and unfortunately when gravity wasn't involved, its movement detection was sketchy.
Decent games like Red Steel suffered from it - horizontal sword movements would completely fail just because the wiimote couldn't effectively pick it up. Then you'd get crap like this happening. Red Steel didn't do so well (with a metacritic score of 55) due to the controls being pretty much broken. After the Wii MotionPlus came out, so did Red Steel 2 and it was far better.
The Wii Remote itself contains an accelerometer, that can detect acceleration on the x-, y-, and z-axes. The WiiMotion Plus adds a gyroscope, that allows for the remote to detect rotation (which would not be considered movement to the accelerometer), and linear movement. Essentially, it makes the Wii Remote more accurate and allows games to better mimic what the player is doing.
Wii Sports Resort is an example of a game that requires the WiiMotion Plus accessory. IIRC you can't play the game without it. Games need to be designed to manipulate the extra data from the accessory, so I don't believe there are any games where using the accessory is optional.