Bob Ross usually starts his paintings with a completely dried base coat. He starts this particular episode explaining that he already prepared a coat of alizarin crimson, but that should make the canvas red, not black.
What I assume he did is:
- Prepare the canvas with a black base coat and let it dry completely
- Right before the show, add a thin layer of alizarin crimson. He most likely did it in the exact same way he shows how to add the vandyke brown, only spread over the entire canvas instead of just the edges. It's vitally important that this layer is very thin, but at the same time covering the entire canvas.
- (This is where the show starts) Add an equally thin layer of vandyke brown to the edges of the canvas and blend it with the crimson.
- Paint the sun with cadnium yellow, moving the brush from the center of the sun to the outside in a circular motion. That leaves a lot of yellow paint in the center that doesn't mix with the crimson + brown base coat. Around the center, the brush picks up more and more of the crimson + brown base and it gets evenly mixed with the yellow, creating a soft transition.
To keep the colors of a painting vibrant, you mustn't mix them with black. So working on a fresh black base coat will inevitably make the colors muddy and brown because the brush always picks up some of the base coat.
If your light colors (white, yellow, etc) still come out muted even after you let the base coat dry, the paint you're using might be low quality. Painting light colors over a dark background requires a lot of pigments to cover up alle the black, but pigments are the most expensive ingredient in paints. So cheap paints usually contain less pigments and more fillers than expensive paints, which makes them less opaque.