As you develop your film in the tank, your film uses up developer chemical components that is TOUCHING the film emulsion first. If you don't agitate, that portion of the chemical gets used up and loses power to develop the film. Slow diffusion takes place so it never completely loses it, but it slows down. When you agitate, you move the chemical around so fresh chemical will be in contact with the emulsion. The developing process speeds up again until the next agitation cycle.
More you agitate and more frequent you agitate, faster developing action takes place because of this.
Now, when you develop film, shadow portion gets done first but highlight portion keeps going longer you process your film. That means, more you agitate and speed up the process, your highlight gets brighter and brighter, thus increasing contrast.
That's the simplest I can make it to explain this.
Now with that said, it is VERY important to keep your process the same every time you process your film. You control your development by adjusting and keeping constant, development time and agitation schedule. If you don't at least keep one the same, keeping consistent result will become very difficult.
Why the tank matter? The amount of chemical is different and how the fluid mix together is different. So Kodak specifies which kind of tank they are talking about when they mention what/how to process.