To get into the business parts of the rheostat you will need to remove the metal clips that holds the the dial and slider components against the coil and coil housing. Just pull outward on the clips and it will come off easily. You might deform the metal clips a little, but you can bend them back in a little to make sure they do their job once reassembled.
Here is what the rheostat looks like on the inside. I am holding on to the dial part that turns when you spin the little knob on the front of the dash panel to adjust the light brightness (same knob that resets the tripometer, by the way).
CAUTION: Do not attempt to take the rheostat apart in your car or in your driveway or anywhere but over a table top where you can watch for the two little springs that are going to fall out. Once you get the dash panel out of your car, I would take it to where you plan to do the work before you even remove the rheostat from the panel. It is not a pain to deal with these springs, but if you lose just one of these its all over for your rheostat (although you can always do the jumper wire trick).
Okay, it's not going to look like this when you pull it apart, the sprigs and slider will fall out. This is just a staged shot to show you how the slider sits on the dial mechanism (the two little springs are currently underneath it - their job is to hold the slider against the coil when everything is assembled. You can see the coil in the housing to the right.
Another staged shot to show you how the springs sit in the dial mechanism. The slider is removed and laying in the background. The metal braket that holds it all together doesn't actually go there - i was just using it to hold the dial upright so I could take the picture.
Here are all the pieces. Do not attempt to remove the coil from the housing as it is attached on one end. Break the coil off and, again, you will be resorting to the jumper method. It's not fragile, just don't pull it out.
The coil will pull up from the side that it is not attached. Many others have noted the white powder material that seems to build up inside (I cleaned it out of the coil area - see it below the housing.) Not entirely sure what that is. May be that the housing is made out of bakelite and it degrades a little over time. Regardless, if your rheostat isn't working you probably have some of the white stuff along with corroded components.
Although the finger projection of the slider runs along the coil (which is fairly clean), there are other contact points of the slider that make contact with the little disc in the center. See how corroded it is? We need to clean that . . . and the slider, too.
